Rogers, who joined Costa as financial director in 2005 before moving up to the top job in 2012, will step down from the board on 20 April, but will stay on until 1 July to help ensure a smooth handover to Dominic Paul, who joins on 6 June.
Despite the succession planning, Whitbread's share price slid 1.8 per cent on the news.
Paul is currently senior vice president of Royal Caribbean International, where he is responsible for the international business outside of the firm's US operations, and has been in the role since November 2013. His focus has been on Asia Pacific and China.
Other roles include managing director and chief operating officer of BMI, and a number of positions at British Airways, Go Fly and EasyJet.
Whitbread chairman Richard Baker praised Rogers for his "immense contribution" to the group over the last 11 years.
"On behalf of all my board colleagues, I should like to thank him for all he has done over the years and, more recently, for his support and typical professionalism in enabling a smooth succession planning process. We wish him every success in the future," he said.
Rogers said:"I have loved being part of Whitbread and Costa during what has been a period of enormous value creation and growth, and after 11 years, I have decided that now is the right time for change."