The Blues have paid an initial £12.5m fee for the 23-year-old, who has signed a seven-year contract at Stamford Bridge.
Chelsea needed a new shot stopper between the sticks after Edouard Mendy moved to the Saudi Pro League and Kepa Arrizabalaga joined Real Madrid on loan.
He will provide competition and cover for Robert Sanchez, who only joined from Brighton and Hove Albion this summer.
"It's a big step for me and it's always been a dream of mine to sign for one of the biggest clubs in the world," Petrovic said.
"Today I have achieved that goal and I'm so happy.
"Playing in the Premier League is something I'm really looking forward to doing and I hope to learn many things here at Chelsea. I can't wait to meet everyone and play at Stamford Bridge."
Petrovic has won two caps for the Serbia national team.
"We are very pleased to welcome Djordje to the club. His performances in MLS have earned deserved praise and recognition -- and have prepared him for the challenge of playing for Chelsea," Chelsea's co-sporting directors Paul Winstanley and Laurence Stewart said:
"Djordje improves an already strong goalkeeping team. We look forward to watching him work under Mauricio Pochettino, goalkeeper coach Toni Jimenez, and the rest of the first-team staff this season."
