Molinari made a 4-foot birdie putt on the second playoff hole to secure the win after both players finished regulation at 13-under 271.
"I played well today but Robert did too," Molinari said. "We pulled away from the rest of the field and had a good battle in the playoff which I was happy to win."
Molinari, who went into the final round with a one-stroke lead over the Swedish veteran, shot a 5-under 66 on Sunday while Karlsson fired a 65.
On the final playoff hole, Karlsson missed a 12-foot birdie attempt, allowing the 28-year-old Molinari to calmly sink 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 shot an even-par 71 to finish in a tie for 20th at 1-under 283.
