Later both Tony Blair and the BBC tried to lower the temperature by promising to "draw a line" under the bitter dispute which sparked the Kelly affair.
But ministers scented blood, with some saying privately that Richard Sambrook, the BBC director of news, should resign or be moved to another post for failing to brief the board of governors properly about Mr Gilligan's report-- which Lord Hutton described as "unfounded" in his report on Wednesday.
At the cabinet's weekly meeting, ministers congratulated Mr Blair on his clear vindication by Lord Hutton. Although Downing Street insisted there was "no gloating," Mr Blair's official spokesman said the cabinet's message was that serious issues such as Iraqi weapons should be discussed without impugning the integrity of politicians.
Some MPs expressed concern that the Government would use its crushing victory to undermine the BBC's editorial independence and put pressure on the rest of the media. Worried by the backlash among BBC staff, ministers moved to calm such fears, promising they would do nothing to put the BBC's independence at risk.
Mr Dyke fell on his sword reluctantly and some friends suggested he had been forced out by BBC governors at a crisis meeting yesterday. He told staff: "I don't want to go.
But if in the end you screw up you have to go." He told the staff who walked out to support him he was not "a political animal" but he hoped the resignations of the BBC's two most senior figures meant "a line can be drawn under this whole episode".
Lord Ryder, a former Tory minister who said he did not want the chairman's job permanently, said: "The BBC must now move forward in the wake of Lord Hutton's report, which highlighted serious defects in the corporation's processes and procedures. On behalf of the BBC I have no hesitation in apologising unreservedly for our errors and to the individuals whose reputations were affected by them."
Mr Blair immediately welcomed the BBC's move, saying: "This for me has always been a very simple matter of an accusation that was a very serious one that was made. It has now been withdrawn, that is all I ever wanted."
He added: "I want to make it absolutely clear I fully respect the independence of the BBC. I have no doubt that the BBC will continue, as it should do, to probe and question the Government in every proper way. What this does now is allow us to draw a line and move on."
Downing Street denied the Government had demanded the head of Mr Davies and Mr Dyke, both Labour supporters. It said: "They decided to resign and the Prime Minister believes that two decent and honourable men have done the decent and honourable thing."
Alastair Campbell, the former Number 10 communications director who was exonerated by Lord Hutton, signalled an end to his personal battle with the BBC.
He told Sky News last night: "I'm content with the fact that finally, after all that everybody's had to go through, that these allegations have been withdrawn. It's for the BBC to decide whether having somebody like Andrew Gilligan on their payroll is a way to restore their integrity and their reputation."
The National Union of Journalists, Mr Gilligan's union, said he wanted to remain at the BBC.
There was continuing surprise in the political and broadcasting worlds that Lord Hutton had come down so heavily on the BBC while acquitting the Government of almost every charge levelled at it during his inquiry.
Sir Christopher Bland, a former BBC chairman, said there was a "curious imbalance" in a report which had exonerated the Government but "tarred and feathered the BBC".
"It is legitimate to question whether Hutton was even-handed in the way he treated on the one hand politicians, civil servants and the security services, and on the other hand the standards of conduct he applied to journalists and broadcasters," he said.
Lord Rees-Mogg, a former BBC vice-chairman, declared: "I don't have any confidence in Hutton.... I have already come to the conclusion that his evidence does not support his conclusions and that it is, put quite simply, a bad bit of work."
An NOP poll for the London Evening Standard found that 49per cent of people believed the Hutton report was a "whitewash", while 40per cent did not. Some 56per cent of people believed Lord Hutton's criticism of the BBC was too harsh.
Michael Portillo, the former cabinet minister tipped as a possible BBC chairman, said the Government had won a "terrific victory" but should stop its "triumphalism" and not use the Hutton report to make the BBC subservient to it.
Number 10 hoped a new chairman of governors - chosen under strict new rules for public appointments - would be in place by Easter. It will then be for the governors to appoint a new director general. Mark Byford has been named as the corporation's acting operational boss.
Nicholas Gardiner, the Oxfordshire coroner, said he was ready to examine statements from witnesses who withheld their evidence from the Hutton inquiry and would ask Thames Valley Police to hand over the missing material. He will then consider Lord Hutton's report and decide within the next month whether to hold a full inquest into Dr Kelly's death.
A number of witnesses took advantage of an option not to have their statements passed to Lord Hutton.
The police interviewed some 500 people and took 300 witness statements during the investigation, passing on fewer than 70 they deemed relevant to Lord Hutton. The coroner's inquest was adjourned under the 1988 Coroners Act, which allowed the inquiry to act as an inquest.
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INDEPENDENT
Greg Dyke, BBC Director General:
Resignation statement
Greg Dyke's e-mail to BBC staff
Lord Hutton:
Report of the inquiry into the circumstances surrounding the death of Dr David Kelly
Herald Feature: Iraq
Iraq links and resources