Entwistle, who only took over as director-general only in September, said he decided to resign taking responsibility for the error as he was also the editor-in-chief.
Tim Davie, who is currently the BBC’s director of audio and music, will take over as acting director general.
Earlier Saturday, Entwistle had said it was “fundamentally wrong” of the current affairs program Newsnight to air an interview with a man claiming he was repeatedly abused by a senior Conservative politician at a children’s home in the 1970s.
The program did not identify the politician in last week’s report, but he was widely named on the Internet as former Tory party treasurer Alistair McAlpine.
McAlpine went public on Friday to strongly deny the allegations, and hours later his accuser Steve Messham retracted his claims, saying he had only just seen a picture of McAlpine and he was not the man who abused him.
Entwistle’s resignation comes just weeks after a storm of allegations that the late BBC presenter Jimmy Savile, one of the corporation’s biggest names, sexually abused hundreds of children over four decades.