The Bosnia international, 28, has been handed the number one jersey, but is expected to be deputy to Thibaut Courtois at Stamford Bridge.
“I am very happy to be joining Chelsea FC,” Begovic told the club’s official website.
“After speaking at length to the manager, I feel like I can develop here and be an important part of this team.”
Begovic established himself as a leading Premier League stopper during a five-and-a-half year stint at the Britannia Stadium.
However, ousting Courtois from the Chelsea goal could prove near-impossible, with Petr Cech recently sealing a switch to Arsenal after finding game time hard to come by at the Blues.
In a statement on his personal Twitter account, Begovic thanked the Stoke staff, players and fans.
“I have loved my time at Stoke, but feel I have to take this opportunity for me to continue to develop and challenge myself as a player,” he wrote.
“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.”
A very, very solid number two to have.
But why on earth would Begovic want to move to Chelsea? He’s not going to play in any meaningful fixture, unless injury or shocking form befalls Courtois. He can look forward to numerous League Cup outings, occasional low-key Champions League matches and numerous medals he played little or no part in earning.
It may be all very exciting for him now, but skip forward to August 8 when Swansea roll into west London and Begovic will begin his bench-warming adventure. Then he might start having second thoughts.