Cohen defended the broadcaster's decision to invest heavily in the rights for the talent show, which will air early next year.
Asked at the MediaGuardian Television Festival why he wanted the show, Cohen said: "I thought it was really good. Every so often the BBC occasionally buys something from abroad that is peerlessly good, like The Apprentice.
"We adapted that for Britain with Alan Sugar and made the British version the best in the world. We hope The Voice will do similarly. We shouldn't shut the door on anything."
Cohen insisted that the BBC would still be investing in British entertainment with the return of Magicians, Let's Dance For Comic Relief and "a brilliant series of Strictly".
He also confirmed that Strictly would not be going head-to-head with The X Factor again, arguing that he was "not really interested" in winning the Saturday night ratings battle with the Simon Cowell show.
'The Voice' would have been in 'X Factor's shadow, says ITV