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The 33-year-old from Michigan was bogey-free in a tournament reduced to 54 holes due to bad weather and birdied the second extra hole on Monday to deny Jamie Lovemark and Byeong-Hun An their first PGA Tour victories.
Stuard rolled in a birdie at the 18th to join Lovemark and An in a play-off and sealed the win by tapping in for another birdie after a superb approach to earn $1.2million prize money and book his place in the 2017 Masters.
Lovemark was left to reflect on what might have been after he had two putts to win it at the 18th to card a 68 at TPC Louisiana, while An was also 15 under through 56 holes following a superb closing 65.
Stuard then came to the fore, landing a magnificent approach shot two feet from the pin and making no mistake to go one better after twice finishing runner-up on the PGA Tour.
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The world number 513, who held a share of the lead with Jhonattan Vegas when Saturday's play was cut short due to more heavy rain, can now look forward to competing at Augusta for the first time next year.
He made birdies on the second and 11th holes and a further crucial gain on the last after completing his second round with a four-under 68.
Bobby Wyatt, the 1,197th-ranked golfer in the world before the tournament, finished 14 under following an impressive closing 64, while Vegas, Jason Day and Chris Kirk were a further shot back in fifth.