Molinari made a 4-foot birdie putt on Sunday to win the $2.2 million tournament after both players finished regulation at 13-under 271.
Molinari shot a 5-under 66 in the final round, while Karlsson had a 65.
Karlsson missed a 12-foot birdie attempt on the second playoff hole and Molinari calmly sank his winner in steady rain at the Phoenix Country Club.
Ireland’s Shane Lowry finished tied for third with Japan’s Hirofumi Miyase at 7-under 277 after both shot 69 in the final round.
Japanese teenager Ryo Ishikawa had a 71 to finish in a tie for 20th at 1-under 283.