Cohen is currently controller of BBC Three, where he has grown the digital channel's audience share in its target 16- to 34-year-old age group by 58% since he took over in May 2007.
Prior to joining the BBC, Cohen was head of E4 and head of factual entertainment at Channel 4, where he commissioned shows such as Skins, The Inbetweeners, Fonejacker and Supernanny.
He has now stepped up to take over Britain's most watched channel, BBC One, as a replacement for Hunt, who is becoming the new chief creative officer of Channel 4.
In his new role, Cohen will join the BBC Vision board, reporting to director of Vision Jana Bennett. He becomes controller of BBC One with immediate effect, but will continue to oversee BBC Three until a successor is appointed.
Announcing his appointment, Bennett said: "Danny Cohen is one of the most talented TV executives of his generation. His work both here at the BBC and at Channel 4, with programmes such as Skins, The Inbetweeners, Being Human, Women, Weddings, War And Me and Young Voters' Question Time attest to his skills as an innovator and a great creative thinker.
"He has a deep understanding of public service broadcasting and a finely developed sense of what audiences of all ages are looking for, things which are hugely important as controller of the UK's most popular channel."
Cohen added: "It's a huge privilege to take charge of Britain's most-watched channel, and I feel honoured to have this opportunity. BBC One is one of the nation's great cultural institutions and I'm looking forward to working with a wonderful range of talented people in the coming years.
"Jay Hunt's success in the job means that the channel is currently in very good shape indeed and I hope to build on these very strong foundations. I'm excited by the challenge."