Chris Patten, the chairman of the BBC Trust, said the crisis had revealed a need for a "thorough, structural, radical overhaul" of the organisation, although he said he would not be quitting.
He was speaking after the dramatic resignation of Entwistle on Saturday night after the flagship BBC news programme Newsnight admitted it had wrongly implicated a politician in child sex abuse.
Entwistle's departure leaves the organisation in chaos as it struggles to restore trust in its journalism and battles the scandal surrounding Jimmy Savile, the late BBC television star now alleged to have been a prolific child sex offender.
"The wholly exceptional events of the past few weeks have led me to conclude that the BBC should appoint a new leader," Entwistle said in a statement outside the broadcaster's London headquarters Saturday.
"To have been the director-general of the BBC even for a short period, and in the most challenging of circumstances, has been a great honour."
The 50-year-old was only appointed to the top job in September, making his leadership is the shortest in the BBC's history.
Entwistle announced his resignation the day after Newsnight was forced to apologise for wrongly implicating a senior Conservative party figure in abuse at a Welsh children's home in the 1970s.
The director-general condemned the report as "fundamentally wrong", although he admitted he had no knowledge of the show before it was aired, a revelation that led critics to suggest he was no longer in control.
Entwistle said quitting was "the honourable thing to do" since he was the BBC's editor-in-chief, and ultimately responsible for all output.
He has been replaced by Tim Davie, a former Pepsi executive who is currently the BBC's director of audio and music, while the BBC Trust finds a more permanent replacement.
Patten said he had made no attempt to persuade Entwistle, whom he had appointed, to stay on. He described him as a "very, very good man -- cerebral, decent, honourable, brave" who was overwhelmed by events.
Patten denied that he should also quit, saying: "I think that I now have to make sure that, in the interests of the licence fee payer and the audience, that the BBC has a grip, that we get ourselves back onto the road."
The BBC has launched several investigations into the Savile scandal and the Newsnight row.
"My job is to make sure that we learn the lessons of those inquiries and that we restore confidence and trust in the BBC," Patten said.
Commentators described the crisis as one of the deepest in the BBC's 90-year history.
The Mail on Sunday condemned Entwistle as the "DG who turned a blind eye to (the) BBC meltdown -- and paid the ultimate price", while the Independent on Sunday's verdict on Entwistle was: "Out of touch. Out of depth. Out of a job".
Entwistle, who first joined the BBC in 1989, was only installed as its director-general on September 17 and had spent his short time at the top dealing with the fall-out from the Savile scandal.
This included a row over why Newsnight dropped an investigation last December into allegations about Savile, amid accusations it clashed with tributes to the late presenter planned for the Christmas period.
Rival broadcaster ITV aired interviews with some of Savile's alleged victims in October, sparking a wave of claims about him.
As director of BBC Vision, Entwistle was in charge of all television output when the Savile report was axed, and he also approved the Christmas tribute.
Entwistle, who edited Newsnight himself a decade ago, admitted on Saturday that the corporation faced a "crisis of trust".
Culture minister Maria Miller welcomed his departure as "regrettable but the right decision", saying: "It is vital that credibility and public trust in this important national institution is restored."
But Newsnight anchor Jeremy Paxman, famed in Britain for his forthright style, said Entwistle had been "brought low by cowards and incompetents".
"I had hoped that George might stay to sort this out. It is a great pity that a talented man has been sacrificed," he said.