Former deputy controller Cooper was named acting controller of the two stations in July after Parfitt announced his intention to move on after 30 years at the corporation.
"I am delighted to announce Ben Cooper as controller of BBC Radio 1 and 1Xtra. It is excellent to have someone of Ben's calibre and experience to lead Radio 1 and keep the station as relevant, popular and distinctive as ever," said director of BBC audio and music Tim Davie.
"Ben has an outstanding track record at Radio 1 and 1Xtra, where he has played an integral role in developing the quality and reach of both networks, and I look forward to working with him in building Radio 1's digital future."
Cooper added: "This is a dream come true. I'm honoured to be leading the BBC's youth music radio stations and care deeply that Radio 1 & 1Xtra lead the way in shaping media consumption for young audiences in the UK.''
Cooper started his career in BBC Local Radio, presenting and producing on Hereford & Worcester, before becoming a producer at Radio 1.
The 41-year-old left the corporation temporarily to join the commercial sector at Capital Radio as an executive producer, before returning to Radio 1 as head of mainstream and then head of programmes.
During his time as deputy controller of Radio 1, Cooper helped implement a major revamp of the schedule, shifting Fearne Cotton to weekday mornings and Greg James to lunchtimes.
He also recently launched Radio 1's Big Conversation, an annual event bringing together young people, politicians and charities to discuss the major issues facing British youth today.
Radio 1 posted its biggest audience since 1999 in the third quarter of 2011, attracting 11.85 million listeners each week, according to the latest RAJAR data.