Five players finished tied for the lead at four-under 23 after the short, nine-hole layout but Ernie Els and Nick Watney opted out of the play-off, leaving just Potter, Mickelson and Kuchar to shoot it out.
Mickelson was eliminated after the first extra hole when he made a par and his two opponents both birdied the hole.
Potter, who is making his debut at the Masters this year, sealed the win and the victor's crystal trophy when he birdied the next hole and Kuchar could only salvage a par.
"Just to be here and be a part of it is awesome and a dream come true," the left-hander said. "To come out here and play in the par-3 and win it is special to me."
Winning the fun-filled nine-hole event, where players are allowed to use friends and family as caddies, has proven to be a bad luck omen for the main Masters tournament, which starts Thursday at Augusta National.
In the 53 years the event has been staged since Sam Snead won the inaugural contest in 1960, no Par-3 winner has gone on to victory in the Masters the same year.
Raymond Floyd came within an eyelash of ending the jinx in 1990, but lost the Masters in a play-off with Nick Faldo.
"I don't care about the curse saying you can't win both," said Potter, who is ranked 140th in the world and was granted a place in the Masters after winning his maiden US Tour title last year. "Just winning something here makes my week already."
Watney aced the ninth hole and Ben Crenshaw made a hole-in-one at the seventh but the biggest roars were reserved for the caddies, who were dressed in white boiler suits.
The caddies ranged from wives, children, grandchildren, parents and partners.
Northern Ireland's Rory McIlroy was accompanied by his girlfriend, Caroline Wozniacki. The Danish tennis player even teed off at the ninth hole but her drive plopped into the lake.