Third-round leader Billy Horschel held a two-shot advantage going into the final 18 holes, but Laird charged ahead with a round that matched the course record at TPC San Antonio.
He birdied the final three holes and claimed his third PGA Tour victory with a 14-under-par 274.
The Scotsman also punched the ticket for his third straight trip to Augusta.
Rory McIlroy mounted a late challenge but ended two off the pace at minus-12 with a final-round 66.
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