His departure leaves England international Jack Butland and recent signing Shay Given as the senior 'keepers at the Britannia Stadium.
In a statement released on his twitter page Begovic said: "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.
"I have loved my time at Stoke, but feel I had the opportunity for me to continue to develop and challenge myself as a player.
"I am very proud of what we have achieved during my time at Stoke, and it has been a great team of players to work with. I wish you all the best for the season ahead."
Begovic has signed a four-year contract and is Chelsea's second summer signing following a loan deal for striker Radamel Falcao.
He told the Chelsea website: "I am very happy to be joining Chelsea. 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."