David Cameron edged the first major TV clash of the general election campaign against Ed Miliband by a margin of 54% to 46%, according to a snap opinion poll released moments after the end of the 90-minute programme.
The Battle for Number 10 show broadcast live on Sky News and Channel 4 saw the Conservative and Labour leaders grilled by veteran TV inquisitor Jeremy Paxman and answer questions from a studio audience.
Mr Cameron urged the electorate to vote for a "strong economy", while his Labour rival insisted: "We can do a lot better than this. We're a great country."
Mr Cameron was faced with questions about his friend Jeremy Clarkson - saying the BBC's decision to sack him was "absolutely right for them" - and admitted that he would not be able to live on the kind of exclusive zero-hours contract that the coalition has outlawed.
Mr Miliband told a studio audience his relationship with brother David was still "healing" after their bruising battle for the Labour leadership, but said he still thought he was the right man for the job.
And he dismissed suggestions that he was a "north London geek" without the steel needed to lead the country, insisting: "Am I tough enough? Hell, yes, I'm tough enough."
In the ICM poll for the Guardian, Mr Cameron narrowly beat Mr Miliband on having the best arguments, seeming more convincing and having the more appealing personality. But out of those who said they may change their minds on which way to vote, 56% said they could move to Labour against just 30% for the Conservatives.
Despite being in the same TV studio for the live programme, the Prime Minister and Labour leader did not go head to head, after Mr Cameron refused to take part in a one-on-one TV debate with the man vying to replace him in 10 Downing Street.
Mr Cameron said the UK was "immeasurably stronger" after five years of his premiership and claimed that "we've turned the economy round". He defended his stewardship of the NHS, after an audience member accused him of breaking his promise not to impose a top-down reorganisation on the health service.
The Conservative leader told his audience in the studio and at home watching TV: "You are going to have to make this huge choice in 42 days' time. What I have learnt in the last five years is that nothing you want to do will work without a strong and growing economy.
"The schools we want for our children, the hospitals we want when we are ill. These things need that strong economy."
Mr Miliband said: "I think this is a choice between those who think this is as good as it gets for Britain and those who think we can do a lot better than this.
"I don't think it's good enough that we've got 700,000 people on zero hours contracts and I think we can do something about it.
"The Prime Minister said he couldn't live on a zero hours contract, well I couldn't either, so let's do something about it."
Asked by an audience member: "You seem gloomy most of the time - are things really so bad?", Miliband replied: "No, but they could be a lot better."
Mr Cameron defended his decision to announce that he would not seek a third term if re-elected in May. He said he was "passionate about having another term" and insisted he would serve "every day" of it but would not be "one of those leaders like Chairman Mao who thinks you can go on and on and on".
Mr Paxman told the PM that many voters found it "problematic" that he had chosen to surround himself with people like Clarkson, ex-HSBC boss Lord Green and former News of the World editor Andy Coulson, asking: "What do you have in common with all these rich people?"
But the PM replied: "The aspersion you are trying to cast is completely ridiculous."
Mr Cameron admitted he had not asked Lord Green about possible tax avoidance in HSBC's Swiss branch at the time of his appointment as a trade minister but said that "all the normal processes and procedures were followed" and that allegations the bank helped clients dodge tax had emerged only "subsequently".
And on Clarkson's dismissal from Top Gear after a fracas with a producer, he said: "I just simply answered a question I was asked and said he is a friend of mine, he is a talent and I hoped it could be resolved. Treating the people you work with badly is not acceptable. The BBC have made their decision and that is absolutely right for them."
Mr Cameron insisted: "What I have done for the last five years is lead a government that has got the economy growing, has got people back to work, has cut the taxes of the poorest people in our country.
"I am not saying we have achieved everything we set out to do, but the country is immeasurably stronger." Some 1.89 million jobs had been created since 2010, and it was a "myth" that all of them were low-paid
He said the Government had outlawed zero-hours contracts with exclusive clauses requiring employees to work only for one company, adding: "I couldn't live on one of those."
Mr Paxman told Mr Miliband that even Labour MPs considered him a "liability" and that he was seen as "a north London geek" by many voters, who thought: "What a shame it's not his brother."
Mr Miliband acknowledged the 2010 Labour leadership battle was "bruising for me, it was bruising for David" but said their relationship was "healed - or healing" and added: "I think I am the right person for the job. I thought it then and I think it now."
The Labour leader insisted the brickbats were "water off a duck's back", adding: "People have to decide - do they want my ideas, do they want my principles when I stood up not just to President Obama, but Rupert Murdoch, the energy companies, the banks.
"I don't care what the newspapers write about me, because what I care about is what happens to the British people, and I know that this country could be so much better."
He said he was "a pretty resilient guy" who had been "underestimated at every turn", telling Mr Paxman: "I know I'm the right man for the job - that's why I'm sitting here and that's why I believe that I'm the best choice to be Prime Minister."
Mr Cameron accepted that his Government had failed to meet his pledge to get net immigration down to the tens of thousands but said it remained "the right ambition".
He said spending cuts planned by the Conservatives were "basically similar" to what had been done in the past five years, and insisted it would be "possible" to make £12 billion in savings from the welfare budget.
Asked if he would allow more private provision of NHS services, Mr Cameron said: "If it's good healthcare, that's what matters to me. I love our NHS. It has done amazing things for my family."
He praised the "amazing" treatment given to his severely ill son Ivan, adding: "I want to make sure that is always there for families in our country. That will always be predominantly an NHS provided by NHS providers."
But the PM faced tough questioning on health from one audience member, who told him: "I saw your billboards - it's safe in our hands, no top-down reorganisation of the NHS. The reality has been very very difficult ... The promises you made last time have been broken, as far as I'm concerned, so how can we trust you next time?"
Mr Cameron responded: "The biggest promise we made about the NHS at a time when we were going to have to make difficult cuts in public spending ... we said, 'We will not cut the NHS'. And we haven't. We've increased spending on the NHS by £12.7 billion over the last five years."
Mr Miliband said that under Labour there would be reductions in spending outside of protected areas such as health and education.
Pressed on the value of the spending cuts, he said: "The figure is at least hundreds of millions, more than a billion pounds.
"That's not the point - we are going to make these decisions when we are in government but ... I have set out an overall approach."
Mr Paxman urged Mr Miliband to make clear what agreement he would make with the SNP, prompting Mr Miliband to renew his pledge not to scrap Trident.
He added: "I'm not going to get into a bargaining game with Alex Salmond."
When Mr Paxman suggested he would indeed have to make a deal, Mr Miliband quipped: "Don't be so presumptuous, there's six weeks to go - you don't get to decide the election result. You're important Jeremy but not that important."
As the interview ended, Mr Paxman appeared not to realise his microphone was still live as he asked Mr Miliband: "Are you OK, Ed? Are you all right." Mr Miliband replied: "Yeah. Are you?"
:: Some 1,123 people who had watched the Battle for Number 10 programme responded to ICM's poll within minutes of its conclusion.