Two key developments led to the Scotsman's first tournament win on the European Tour.
He sank a 30-foot birdie putt on the first playoff hole, and on the second, Garrido's approach shot to the green went into the water and he was finished.
Both players ended regulation with scores of 15-under-par 273. Garrido had held a three-stroke lead when the round began. Lawrie shot 5-under 67 in the final round, while Garrido managed an even-par 72.
"I knew if I got things going today, I would have a decent chance of winning," Lawrie said. "I've been playing quite well lately without really getting my rewards, so maybe it was just my turn today."
Denmark's Soren Hansen (69), who missed a 10-foot birdie putt at No. 18 and did not qualify for the playoff, finished alone in third place at 14-under 274.