Potter seemed like a lock to take his first PGA Tour title on extra hole No. 2 when he chipped to within 4 feet of the pin on the par-three 18th at The Old White TPC, but his birdie try ran around the edge of the cup.
Kelly, meanwhile, came out of a green-side bunker to salvage a par, setting up a third extra hole.
Again, Potter's chip off the tee was an excellent one, putting him within 4 feet of the hole. After Kelly was wide on a long birdie putt try, Potter made sure he didn't miss again, draining his shot for the tourney-winner.
Potter had fired a 6-under-par 64 in the final round, ending at 16-under 264, knotted with Kelly, who carded a 66 in regulation.
Sharing third place at 14-under were Charlie Wi (65) and Charlie Beljan (67).
Daniel Summerhays (64) ended alone in fifth one shot further back at minus-13.