International Trade Secretary Liam Fox said the Government is "disappointed" at the High Court ruling that Theresa May cannot trigger the formal process of leaving the European Union without the prior approval of Parliament.
Speaking in the House of Commons, Mr Fox said "the Government is determined to respect the result of the referendum".
"The Government is disappointed by the court's judgement," he said.
"The country voted to leave the European Union in a referendum approved by Acts of Parliament. The Government is determined to respect the result of the referendum.
"This judgment raises important and complex matters of law and it's right that we consider it carefully before deciding how to proceed."
He added: "I have nothing to add other than to reiterate that it's right that the Government will consider carefully before deciding how to proceed following the judgment."
In one of the most important constitutional cases in generations, three senior judges ruled that the Prime Minister cannot use the royal prerogative to trigger Article 50 of the Lisbon Treaty to start the UK's exit from the EU without the prior authority of Parliament.
Scotland's First Minister Nicola Sturgeon, who is lobbying to keep Britain in the European single market and against a so-called "hard Brexit", described the result as "significant indeed".
Interim Ukip leader Nigel Farage said he was worried that politicians were attempting to block or delay Brexit and warned that such a move would provoke huge public anger.
"I worry that a betrayal may be near at hand," he said. "Last night at the Spectator Parliamentary Awards I had a distinct feeling that our political class, who were out in force, do not accept the 23rd of June referendum result."
"I now fear that every attempt will be made to block or delay the triggering of Article 50. If this is so, they have no idea of the level of public anger they will provoke."
Liberal Democrat leader Tim Farron urged the Government to now set out its negotiating strategy to Parliament before triggering Article 50, which Mrs May has promised to do before the end of March.
The PM has insisted that she cannot reveal her hand before beginning negotiations.
But Mr Farron said: "It is disappointing that this Government was so intent on undermining parliamentary sovereignty and democratic process that they forced this decision to be made in the court, but I welcome the news today that MPs will get to vote on the triggering of Article 50.
"Given the strict two-year timetable of exiting the EU once Article 50 is triggered, it is critical that the Government now lay out their negotiating strategy to Parliament, before such a vote is held.
"So far May's team have been all over the place when it comes to prioritising what is best for Britain, and it's time they pull their socks up and start taking this seriously.
"Ultimately, the British people voted for a departure but not for a destination, which is why what really matters is allowing them to vote again on the final deal, giving them the chance to say no to an irresponsible hard Brexit that risks our economy and our jobs."
Leave campaigner Dominic Raab described the decision as "disappointing".
The Tory former minister said: "This case is a plain attempt to block Brexit by people who are out of touch with the country and refuse to accept the result.
"However, the vote to leave the EU was clear and they should not seek to obstruct it.
"Leaving the EU provides us with the opportunity to create a society which works for all. Instead of trying to row back on the referendum result, the country should be moving forward and working together to make a success of Brexit."
A flurry of excitement swept through the House of Commons chamber as news of the court's judgment spread.
Former health minister Ben Bradshaw briefly put his hands up in the air with glee as he heard the result, while Labour MP Kevin Brennan demanded to know if the Government will "respect the ruling of the court".
But pro-Brexit MPs were quick to denounce the decision, with Michael Fabricant (Lichfield) urging ministers to "deplore" it, as he does.
Green Party co-leader Caroline Lucas said the ruling meant Mrs May and her Cabinet would not have "total control" over Britain's future.
"We welcome this ruling, which shows that ministers do not have the power to trigger Article 50 without consulting Parliament," she said.
"Parliament must have the opportunity to debate and vote on triggering Article 50, rather than a group of ministers at the top table having total control over this country's future place in the world.
"The Green Party will continue to fight to protect free movement, workers' rights and the vital environmental protections we currently have as part of the EU."
Labour MP and constitutional expert Graham Allen said the Supreme Court now faces "its first historic test" in hearing the Government's appeal against the ruling.
He said: "It is the beginning of defining more clearly and honestly a separation of powers in the UK, which has hitherto been shrouded in mystery.
"Parliament can no longer be the poodle of Government of any political complexion.
"On fundamental matters of our democracy, Parliament must not only be consulted but, as on Article 50, legislate.
"This is not to overturn the decision in principle by the British people but to give it full life.
"I welcome the decision of the High Court and now hope the Supreme Court upholds the sovereignty of Parliament which was such a core part of the reason to leave the EU."
Former Liberal Democrat leader Lord (Menzies) Campbell of Pittenweem said the ruling was a "slap in the face" for Mrs May.
"This is a clear illustration of the well-known legal principle that no matter how high you are, the law is above you," he said.
"It is a slap in the face for the Government. It shows the dangers of playing ducks and drakes with the constitution and particularly the sovereignty of Parliament."
Former education secretary Nicky Morgan said she hoped the chances of the Government hitting Mrs May's Article 50 deadline were "quite strong".
She told the BBC's Victoria Derbyshire programme: "The triggering of Article 50 is a big first step but I think people will want progress made on that.
"I think it's important the Government shows there is progress and I see no reason why, actually, unless we end up with some lengthy legal proceedings tying things up, the Government doesn't take charge of this and say we are going to have a piece of very simple legislation.
"Members of Parliament will be mindful of how their constituents voted on 23 June."
Shadow international trade secretary Barry Gardiner asked Dr Fox in the Commons: "Will you acknowledge that the vast majority of members in this House are now committed to honouring the decision to leave the EU but that democracy demands that the terms of our leaving must be subject to the proper advanced scrutiny and consent of this democratically elected House and not negotiated in secret and smuggled through without the support of this sovereign Parliament?"
Mr Bradshaw urged the Government to delay the planned triggering of Article 50 in the wake of the ruling.
He said: "Eighty percent of the fish caught around our coastline in the South West goes straight for export to the rest of the European Union and there is huge concern in the industry about the impact of tariffs if we leave the single market.
"Given the excellent news from the High Court and those concerns, wouldn't the Government be wise not to invoke Article 50 as early as March?"
Meanwhile, Tory former minister Anna Soubry urged Dr Fox to examine the details of the ruling.
She said: "I think you said the Government will be appealing. Could I ask you on behalf of the Government please to look at their learned judges' ruling and understand that it's not about a re-run of the arguments about the EU referendum, it's all about Parliament's sovereignty.
"In that event will the Government look at it carefully to decide whether or not in fact the learned judges are right and it's this place that should and indeed will trigger Article 50?"
Dr Fox replied to Mrs Soubry: "As this is now an ongoing court case I've nothing further to add to the comments I made earlier."
Ukip leadership hopeful Suzanne Evans condemned the "activist judges" and suggested they should be sacked.
"How dare these activist judges attempt to overturn our will? It's a power grab & undermines democracy," she wrote on Twitter. "Time we had the right to sack them."
She added: "Predictably, the same people now quoting 'parliamentary sovereignty' are the very same people who were happy to give it away for last 40yrs."
Commons leader David Lidington told MPs: "I can confirm to the House that it is the Government's intention to appeal against today's judgment from the High Court.
"We are, as the House is aware, in a situation where we have this judgment today and a very little while ago a judgment from the High Court of Northern Ireland which came to a completely different decision on the same subject.
"So we now have the High Courts in two different parts of the United Kingdom coming to opposite conclusions on the same constitutional and legal question. So this will need to go to a higher court."
He said the Government will give a statement to the Commons on Monday about the ruling and the appeal.
Ukip donor Arron Banks, co-chairman of the Leave.EU campaign, said unelected judges had declared war on British democracy.
He said: " Parliament voted six-to-one in favour of letting the people decide. They didn't get the answer they wanted, and now they're going to use every dirty trick in the book to try to sabotage, delay or water down Brexit.
"It's no surprise that the legal establishment has joined the political class in declaring war on British democracy. Why wouldn't unelected judges want to preserve an EU system where unelected elites like themselves are all-powerful?
"Tony Blair let the cat out of the bag last week: the Remain campaign hasn't gone away; they don't have any respect for the 17.4 million and they will do everything in their power to reverse the public's historic victory over the Establishment.
"Well, I don't think the people are going to take this lying down. Leave.EU will now be going back into full campaigning mode, and I would urge anyone who believes in democracy to sign up and join the fight."
Wales First Minister Carwyn Jones said it was a mistake for the Government to appeal against the ruling - and repeated his view that the devolved administrations should also get a vote on Mrs May's Brexit negotiating position.
"The High Court ruling is extremely clear - the UK Government cannot trigger Article 50 using the Crown prerogative," he said.
"Indeed, this is consistent with many of the arguments made by the Leave campaign themselves about Parliamentary sovereignty. It is a mistake, in my view, to challenge such a clear ruling and we should now try and move ahead to develop a fuller understanding of the UK Government's position.
"The position of the Welsh Government has been consistent throughout - we accept the decision made by the people and will not work against the referendum result. We are working hard to get the best possible exit terms for Wales.
"However, it is important that votes take place in all four nations to endorse the UK negotiating position."
Leave campaigner John Baron said the decision "must not be allowed to frustrate the will of the British people".
"It was made clear by the Government at the time that the referendum result would be respected by the political establishment. David Cameron and Jeremy Corbyn both said they would invoke Article 50 the day after a Brexit result. People voted in the referendum on this basis. We will man the barricades if the result is not respected."
The SDLP's Mark Durkan, a member of the Brexit Select Committee, said: " The SDLP has been consistent in its objection to the use and abuse of the dubious royal prerogative powers. Those of us who believe in parliamentary democracy know that major decisions where substantive details and strategic options count, should be debated and voted upon by MPs, rather than being pocketed as a private or personal tactic for a Prime Minister."
Richard Tice, co-chairman of Brexit pressure group Leave Means Leave, said: "This is disgraceful - 17.4 million people will be furious today.
"Our democracy is being damaged by an elite band of people in the legal system."
He added: "A vote in Parliament is wholly unnecessary, time-consuming and betrays the democratic will of the people."
Ukip's only MP, Douglas Carswell, hit out at the "shocking judicial activism", adding that "these judges are politicians without accountability".
"I hope reform of judicial appointments is in a manifesto or two," the Clacton MP said.
Writing on Twitter he said: " Remainers, aided and abetted by lawyers, seeking to tell electorate to sod off."
Tom Elliott, MP for Fermanagh and South Tyrone, said the court had made a "surprising" ruling but added that he does not think it will have an effect on Britain leaving Europe.
Speaking outside Westminster, the UUP representative, who voted to leave, said: "I don't actually see the MPs voting against the triggering of Article 50.
"I think the MPs will take a practical decision, they will take a sensible decision that 'Look, we handed this over to the people at a referendum. It's really their decision', and that's what I see happening."
Brexit select committee chairman Hilary Benn said he would vote in favour of triggering Article 50 but it was important that Parliament should have a role in the process.
Senior Labour MP Mr Benn told BBC2's Daily Politics: "This is an important judgment on the sovereignty of Parliament. It is not about a decision on whether we are leaving the European Union, because we are - the British people made that choice in the referendum and I think Parliament should uphold it.
"But there is ... a debate to be had about what kind of relationship we are going to have and therefore the Government's negotiating objectives are really important."
Mr Benn said he would vote in favour of triggering Article 50 "but that is separate from what we should be seeking to negotiate in that process".
Former Remain campaign chief Will Straw said: " Brexit still means Brexit. But sovereignty means sovereignty too."
MEP Janice Atkinson, who used to be a member of Ukip, called for the reinstatement of Mr Farage as permanent leader of the party after the court's decision.
Ms Atkinson said: "To seek now to set Parliament against the will of the people in this way is dangerous, unethical and risks a clash between citizens and Government.
"Ukip now needs to get its act together again fast and ensure that Nigel Farage is back in the driving seat when, once again, the country needs his leadership to ensure our future as an independent nation state. That, and nothing less, is what the British people voted for on June 23."
Shadow Brexit secretary Sir Keir Starmer said the Government must now produce its negotiating plan for MPs to vote on.
"The Prime Minister was trying to sideline Parliament and the court has now ruled against her and reminded her that Parliament is sovereign, so the Government needs to urgently review its approach and refer this issue to Parliament where it should be debated," he said.
Sir Keir told BBC Radio 4's World At One: "Of course there is a mandate for leaving the EU, and we have to accept and respect the result of the referendum.
"But the terms, and how we leave the EU, are vitally important."
He added: "I really don't think that, in light of this ruling, the Government can get away with refusing to disclose its negotiating stance."
Nicola Sturgeon, speaking at First Minister's Questions at Holyrood, said: "The judgment this morning I don't think is a huge surprise for anyone that followed the case but it is hugely significant and it underlines the total chaos and confusion at the heart of the UK Government.
"We should remember that their refusal to allow a vote in the House of Commons is not some matter of high constitutional principle, it's because they don't have a coherent position and they know that if they take their case to the House of Commons that will be exposed."
Asked by Labour MSP Lewis Macdonald if the Scottish Government would "actively oppose" the appeal by the UK Government when the case reaches the Supreme Court, Ms Sturgeon said: "We will be looking at the judgment very carefully and, yes, we will actively consider whether there is a case for the Scottish Government to become participants in that case."
She added: "The job of this government is to protect Scotland's interests. Scotland voted to remain in the EU and my job is therefore to protect our place in Europe and in the single market as far as I possibly can.
"SNP MPs in the House of Commons will certainly not vote for anything that undermines the will or the interest of the Scottish people."
Hilary Benn, chairman of the Brexit Select Committee, described the judgment as "very significant", but said it is not a decision about leaving the EU, adding: "That choice has been made by the British people and I think Parliament must respect it."
But he said it is "perfectly legitimate" for Parliament to ask the Government what the plan for Brexit is.
He said: "I think it's right and proper that the Government should come to the House of Commons to tell us what it is going to be looking for in these very important negotiations that will start next year."
Tory MP Peter Bone said he felt "upset" by the ruling, adding: "I think it's strange that the court is trying to tell Parliament what to do."
He said: "I think that all that's happening with the legal side is you're making a lot of lawyers richer and that's never a good thing.
"What we should actually want to do is get on with Brexit.
"Now if we had a vote in Parliament, I'm sure Parliament would support triggering Article 50."
Vote Leave campaign director Dominic Cummings said that although the judgment had "some bad bits", Brexit supporters should acknowledge that "at heart it is reasonable".
Parliamentary sovereignty means the Government cannot and should not try to change domestic law by use of the royal prerogative, he said.
He urged Brexit-supporting MPs to stick to the "important principles of how a serious country works" and focus on winning the battles in Parliament, rather than "confused whining".
Former chancellor George Osborne joined calls for the Government to set out its plan for Britain's future trading relationship with the EU.
He told Sky News: "The question is not who makes the decision on whether we're leaving the European Union because the public have made that decision in the referendum.
"I think the sooner we get on to the substantive discussion in Parliament with the Government about the nature of our new relationship with Europe, how we're going to trade with these key partners of ours like Germany and France, how businesses here in Manchester and across the country are going to have the certainty so that they can invest and hire people over the next couple of years.
"For me these are the key issues and I would be arguing for the closest possible trading and economic relationship with Europe, outside of the EU, because we've got to respect the referendum."
Pro-EU Tory grandee Ken Clarke said the Government will have to give MPs a vote on its Brexit strategy in a return to "proper parliamentary democracy" rather than decisions being taken in Cabinet committees of "squabbling colleagues".
He told BBC Radio 4's PM: "I think the argument that somehow everything has been decided by this one vote is wrong, it's a childish attempt to avoid debate, the vote was we leave the EU, but it was silent on the terms.
"Even the leading campaigners, the leading Brexiteers do not agree with each other (on the terms)."
Former Tory MP Zac Goldsmith warned that if Parliament tried to block Brexit it would lead to the rise of the far right in Britain.
Mr Goldsmith, who quit his seat to force a by-election which he is fighting on a ticket opposed to Heathrow Airport expansion, told LBC: "If we were to overturn the outcome of the biggest democratic exercise we have ever had in this country - 17 million people - if we put our fingers up to them, if we turn our backs on them, or ignored them, I think we would see the emergence in this country of the kind of far-right extreme movements that plague the continent, but fortunately do not plague this country."