The 39-year-old Australian piled up nine birdies and an eagle at the par-five 12th, and needed only 23 putts as he finished with a 22-under total of 258 on the Old White Course.
"Have I putted as well? Probably not. Today was a purple patch ... By no means do I do this all the time. I had a lot of opportunities, and I made them."
Asked if he felt any extra pressure coming down the stretch with a 59 on the line, he replied: "I felt pretty comfortable, pretty relaxed. It's not a nerve-racking thing to be involved in. It's just an opportunity.
American Paul Goydos was the most recent player to card a 59, in the opening round of last month's John Deere Classic.
Al Geiberger posted the U.S. Tour's first 59 at Colonial Country Club in 1977. The second was registered by Chip Beck at the 1991 Las Vegas Invitational and David Duval matched the feat in the final round of the 1999 Bob Hope Classic.
Appleby, chasing his first victory on the PGA Tour since the 2006 Houston Open, charged into contention with a sizzling outward nine of six-under 28 to lie one stroke off the lead.
However, the Australian again drew level at the 16th before sinking birdie putts from 10 feet on 17 and at the last to seal victory with the most memorable round of his career.
"It is great to do that to win a tournament," Appleby said of his birdie putt on the 18th green. "Not that that was the last hole (of the tournament), but to do that and cap off a low number was very unique."
"A flood of emotions certainly came across me minutes later, what I had done potentially, but what I still had to do," Appleby said. "I had to step away, get back on the range, feel like the golfer again for potentially a playoff.
Overton, whose victory bid stalled with three three-putts, recorded his third runner-up finish of the season.
"I got beat by a 59," the American said. "What can you say? I played great, hit a lot of great shots. You can't win golf tournaments when you putt it that bad."
(Writing by Mark Lamport-Stokes in Los Angeles; Editing by Dave Thompson/Ian Ransom)