The Bosnia-Hercegovina international joins Chelsea on a four-year deal.
He replaces Stamford Bridge legend, Petr Cech, who signed for Arsenal last month after losing the goalkeeping top spot to 23-year-old Belgian Thibaut Courtois last season.
“I have loved my time at Stoke but feel I had to take this opportunity to continue to develop and challenge myself as a player,” Begovic said.
Begovic, who joined the Potters from Portsmouth in 2010, is the champions’ second signing of the summer following the arrival of Radamel Falcao on a season-long loan.
“I am very happy to be joining Chelsea,” said Begovic.
“After speaking at length to the manager, I feel like I can develop here and be an important part of this team. I am looking forward to meeting up with the team on Wednesday for the pre-season tour.”
Current Stoke manager Mark Hughes and former boss Tony Pulis were among the names he thanked for his development.
“I would like to say thank you to everyone at Stoke Football Club, the chairman and Coates family, Mark Hughes and Tony Pulis, goalkeeper coach Andy Quy, all my team mates and especially the fans who have made me and my family so welcome,” Begovic wrote in a post on Twitter.
“I have loved my time at Stoke, but feel I had to take this opportunity for me to continue to develop and challenge myself as a player”.