George Entwistle has resigned as director-general of the BBC after only 54 days in the job, as the crisis in the state-funded broadcaster over child sexual abuse allegations threatens to spiral out of control.

The resignation comes days after the BBC's Newsnight programme was forced to issue an "unreserved" apology to the former Conservative Party Treasurer, Lord McAlpine, whom the programme had falsely implicated in the sexual abuse of residents at a Welsh children's home in the late 1970s and early 1980s.

"In the light of the fact that the director-general is also the editor-in-chief and ultimately responsible for all content," Entwistle said in a statement outside Broadcasting House. "And in the light of the unacceptable journalistic standards of the Newsnight film broadcast on Friday 2 November, I have decided that the honourable thing to do is to step down from the post of director-general.

"The wholly exceptional events of the past few weeks have led me to conclude that the BBC should appoint a new leader. 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 broadcaster has been reeling from explosive allegations that the same Newsnight programme had shelved a report alleging that the late BBC celebrity Jimmy Savile had systematically abused children during his time as a DJ and presenter within the organisation.

Entwistle, who at the time the Newsnight segment was being prepared was also planning a tribute programme to Savile for the peak Christmas viewing period, faced serious criticism for his handling of both that affair and the Newsnight allegations against McAlpine.

Entwistle claimed he was not aware of Newsnight's plans to screen the second investigation, which implicated McAlpine. The allegations were made by a former resident of the home, Steve Messham. McAlpine was identified to journalists by Messham, but not named in the Newsnight film, and therefore not given a right of reply. McAlpine's name surfaced soon after the broadcast on social media sites such as Twitter.

When asked on BBC Radio 4's Today programme why he had not seen the programme, Entwistle said he was "out" at the time.

He also courted controversy when he said he had not read the Guardian newspaper report stating that Messham may have mistakenly identified Lord McAlpine as his alleged abuser.

Last month, during a parliamentary hearing into the Savile affair, MPs were scathing over the editor-in-chief's "extraordinary lack of curiosity" about the Newsnight investigation into Savile last December.

"It's a regrettable situation, but the right decision," said Secretary of State for Culture, Media and Sport, Maria Miller, in a statement. "It is vital that credibility and public trust in this important national institution is restored. It is now crucial that the BBC puts the systems in place to ensure it can make first class news and current affairs programmes."

"This is undoubtedly one of the saddest evenings in my time in public life," said Lord Patten, the BBC Chairman, standing next to Entwistle outside Broadcasting House. "At the heart of the BBC is its role as a trusted global news organisation, as editor-in-chief, George has very honourably offered his resignation because of the unacceptable mistakes and the unacceptable shoddy journalism which has caused so much controversy."

Jeremy Paxman, Newsnight's best-known presenter and one of the BBC's most recognised journalists, issued a statement on the social networking website Twitter via his agent, Capel & Land, that appeared to attack his colleagues and senior managers.

"George Entwistle's departure is a great shame. He has been brought low by cowards and incompetents. The real problem here is the BBC's decision, in the wake of the Hutton Inquiry, to play safe by appointing biddable people.

"They then compounded the problem by enforcing a series of cuts on programme budgets, while bloating the management. That is how you arrive at the current mess on Newsnight. I very much doubt the problem is unique to that programme.

"I had hoped that George might stay to sort this out. It is a great pity that a talented man has been sacrificed while time-servers prosper."

Tim Davie, who has been heading BBC Worldwide since 19 October, will stand in as acting director-general until a permanent replacement can be found.

Click here to read the full transcript of the John Humphrys Radio 4 interview that made it clear that Entwistle had to go.