Cotton will take the 10am to 12.45pm slot, while Whiley will move to weekend afternoons from 1pm to 4pm as part of a major reshuffle of the station's schedule.
The successful Live Lounge, where musicians perform their own and other artists' songs live, will continue in Cotton's new programme.
The 27-year-old said: "I love being at Radio 1 and taking on a weekday show is a dream come true. Jo is leaving very big shoes for me to fill and it’s nerve-wracking.
"She has been a massive inspiration to me throughout my career so I have a high standard to live up to."
Reggie Yates, who currently presents the 4pm to 7pm weekend show with Cotton, will continue to host the programmes on his own.
Whiley said of her move to weekends: "After eight wonderful years, the time has come to move on and give someone else the privilege of entertaining the nation on a daily basis.
"I’ve had the most amazing time ever! The prospect of lie-ins, taking my kids to school and doing something new is, I confess, rather exciting."
Of Cotton, she added: "I think she's fantastic and I couldn't leave the show in a better pair of hands."
Elsewhere, Greg James will host the 1pm to 4pm weekday slot in place of Edith Bowman, who will present weekend breakfast from 7am to 10am. James's early morning breakfast show will now be hosted by Dev.
The changes will start on September 21.