Petrovic has signed a contract that will keep him at the Vitality Stadium until June 2030.
The 25-year-old had been with Chelsea since moving from the MLS club New England Revolution in 2023.
Petrovic, who spent last season on loan at the French team Strasbourg, made 31 appearances for the west London club.
"I'm very pleased to be here. I came to Bournemouth because I want to develop and compete at the highest level," said the Serbia international.
"With this club and its facilities, I believe we can achieve our goals. I aim to assist the team in securing results, improve daily, and become a better player."
Last season, Bournemouth had Kepa Arrizabalaga on loan from Chelsea.
The Spanish keeper has since moved to Arsenal on a permanent transfer, while the Cherries' backup Mark Travers transferred to Everton on Tuesday.
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