Petrovic agreed a contract that will keep him at the Vitality Stadium until June 2030.
The 25-year-old had been at Chelsea since joining from MLS side New England Revolution in 2023.
Petrovic, who spent last season on loan at French team Strasbourg, made 31 appearances for the west London club.
"I'm really happy to be here. I came to Bournemouth because I want to grow and I want to play at the best level," the Serbia international said.
"Together with this club, with these facilities, I think we can achieve it. I want to help the team get the results, improve every day and be a better player."
Bournemouth had Kepa Arrizabalaga on loan from Chelsea last season.
The Spain keeper has since joined Arsenal on a permanent basis, while Cherries' back-up Mark Travers signed for Everton on Tuesday.