Prime Minister David Cameron told the Commons that the inquiry said none of the casualties posed any threat to British troops.
He said the inquiry found that the first shots were fired by British troops, no warnings were given, and some of the soldiers lost control.
The 14 civilians died after British troops opened fire on a civil rights march in Londonderry on January 30 1972.
Mr Cameron told MPs: "What happened on Bloody Sunday was both unjustified and unjustifiable. It was wrong."
He added that ""what happened should never have happened".
"The Government is ultimately responsible for the conduct of the armed forces. And for that, on behalf of the Government, indeed on behalf of our country, I am deeply sorry."
The inquiry found that the soldiers of the support company who went into the Bogside, where the march was taking place, did so "as a result of an order which should not have been given" by their commander.
It concluded that "on balance" the first shot in the vicinity of the march was fired by British soldiers.
None of the casualties was carrying a firearm and while there was some shooting by republican paramilitaries, "none of this firing provided any justification for the shooting of civilian casualties".
In no case was any warning given by the soldiers before opening fire and the support company "reacted by losing their self-control ... forgetting or ignoring their instructions and training".
The result was a "serious and widespread loss of fire discipline".
Afterwards, many of the soldiers involved "knowingly put forward false accounts in order to seek to justify their firing".
The inquiry found that some of those who were killed or injured were clearly fleeing or going to the assistance of others who were dying.
The report's detailed review of the events of January 30 1972 in Derry catalogued scenes of horror that included the image of unarmed victims shot dead as they tried to crawl away.
The key finding were:
"The firing by soldiers of 1 Para caused the deaths of 13 people and injury to a similar number, none of whom was posing a threat of causing death or serious injury." This also applied to the 14th victim, who died later from injuries;
"Despite the contrary evidence given by soldiers, we have concluded that none of them fired in response to attacks or threatened attacks by nail or petrol bombers." The report added that no one threw, or threatened to throw, nail or petrol bombs at soldiers;
The accounts of soldiers were rejected, with a number said to have "knowingly put forward false accounts";
Members of the official IRA fired a number of shots, though it was concluded it was the paratroopers who shot first on Bloody Sunday;
Northern Ireland Deputy First Minister Martin McGuinness, second in command of the provisional IRA in Derry in 1972, was "probably armed with a Thompson submachine gun", and though it is possible he fired the weapon, this cannot be proved. The report concluded: "He did not engage in any activity that provided any of the soldiers with any justification for opening fire."
Lord Saville concluded the commander of land forces in Northern Ireland, Major General Robert Ford, would have been aware that the Parachute Regiment had a reputation for using excessive force. But he would not have believed there was a risk of paratroopers firing unjustifiably;
The commanding officer of the paratroopers, Lieutenant Colonel Derek Wilford, disobeyed an order from a superior officer not to enter troops into the nationalist Bogside estate;
Lord Saville found his superior officer, Brigadier Patrick MacLellan, held no blame for the shootings as if he had known what Col Wilford was intending, he might well have called it off;
No blame was placed on the organisers of the march, the Northern Ireland Civil Rights Association;
Neither the UK nor Northern Ireland governments planned or foresaw the use of unnecessary lethal force.
The report referred to one person who was shot while "crawling ... away from the soldiers" and another who was probably shot "when he was lying mortally wounded on the ground".
A father was "hit and injured by Army gunfire after he had gone to ... tend his son," the report said.
"The immediate responsibility for the deaths and injuries on Bloody Sunday lies with those members of the support company whose unjustifiable firing was the cause of those deaths and injuries," the report said.
It added that "none of the casualties was posing a threat of causing death or serious injury, or indeed was doing anything else that could on any view justify their shooting".
Mr Cameron said: "These are shocking conclusions to read and shocking words to have to say. But you do not defend the British Army by defending the indefensible.
"There is no point trying to soften or equivocate what is in the report. It is clear from the tribunal's authoritative conclusions that the events of Bloody Sunday were in no way justified."
Memories of the Bloody Sunday dead boomed out across Derry's Guildhall Square as bereaved relatives read out the names of their loved ones to an audience of thousands who had packed into the space in front of the city's historic walls.
As each relative in turn came to the microphone on the steps of the venue, they read out a name of one of those killed by the Paras, before shouting the word: "Innocent."
John Kelly, whose 17-year-old brother Michael was found by the report to have been shot by soldiers without justification, made an emotional address to the crowd that recalled the civil rights movement of the 1960s.
"We have overcome," he declared, prompting cheers from the throng.
He said the report had vindicated the families and it would now be the verdict of history for all time.
Mr Kelly produced a copy of the shamed Widgery report, which had largely exonerated the soldiers only months after the killings.
He said the whitewash had been laid bare, before ripping the Widgery document into pieces.