Begovic enjoyed a memorable season with the west London club having won the Premier League title, but failed to establish a regular place in the side as Thibaut Courtois has established himself as first-choice keeper at Stamford Bridge.
Bournemouth were hoping to clinch his signature and the Bosnia and Herzegovina keeper was thrilled to put pen to paper on a long-term deal. He will join up with the south coast club on July 1.
Begovic enjoyed a brief loan spell with Bournemouth back in 2007 where he made eight appearances and is relishing the chance to work under Eddie Howe next season.
Begovic said: ““AFC Bournemouth is an exciting, ambitious club which has everything in place for a very bright future.
“Being part of a title-winning team at Chelsea was amazing but I felt now was the right time to move on because I want to be playing regular football.
“I am coming into an important part of my career and I want to show people what I am capable of.
“When I spoke to Eddie Howe I knew AFC Bournemouth was the right club for me and I want to bring my experience to the squad, both on and off the pitch.
“I’m really excited to be part of the journey here and hope I can help take the club to another level.”
Bournemouth manager Eddie Howe said: “Asmir is an outstanding goalkeeper who has proved himself capable of performing at the highest level over a number of years.
“He is at the right age and the right fit. He is a leader as well and exactly the kind of character you want in the team.
“Asmir will provide great competition to the excellent group of goalkeepers we already have here and his experience will prove invaluable in the dressing room and on the pitch.
“I look forward to him being a very important player here for years to come.”