Allen has held the equivalent post on Channel 4 since 2009, and will now replace Cheryl Taylor, who moved to become the controller of CBBC in June.
While at Channel 4, Allen was responsible for the development of programmes such as Friday Night Dinner, Black Mirror, Facejacker, Derek and Frankie Boyle's Tramadol Nights, as well as two seasons of The Inbetweeners. He also acted as executive producer on The Inbetweeners Movie.
Allen said that this was a "perfect time" to join the BBC as "Cheryl Taylor leaves behind a string of big audience delivering hit shows across all the channels, and [new Director General of the BBC] George Entwistle is a very keen and inspiring champion of comedy on his new watch".
Allen added: "Comedy is my life's passion, so this is a dream job and the best gig in TV comedy given the BBC's unique volume, range, commitment and legacy."
Roger Mosey, acting director of BBC Vision, said of Allen: "He has a great track record in the industry and we're pleased he's joining the BBC to make sure that we continue to offer original and distinctive comedy to our television audiences."