The 31-year-old carded a four-under 67 to finish seven under with Haas, who was one ahead overnight.
He then claimed the play-off with a par for his 15th professional win, but only his second on the PGA Tour.
World number one Jordan Spieth, the defending champion, birdied the last for a 73 and was tied 18th, seven back.
Playing partner Lee McCoy, the 22-year-old amateur who grew up next to the Copperhead course, followed his third-round 66 with four birdies in a 69 and finished fourth to qualify for the Houston Open.
