'I thought when I first won I'd be jumping to the moon,' said Short, who pocketed $720,000 for the victory. 'I know what those guys are talking about when they don't have any emotion. I guess it hasn't set in yet.'
Short needed a birdie at the last to get to minus-21 and he did just that. He ran home a 9-footer on the 72nd hole to force the extra session, his first on the PGA Tour.
They went back to 18 and Furyk had a chance to win the title. He missed an 18-footer for birdie so it was on to the par-3 17th for the second playoff hole.
Furyk hit a 5-iron into the water and would need help to extend the playoff. Short played a 4-iron into a bunker. After his drop, Furyk pitched to 7 feet and watched Short in the bunker.
Short hit a strong shot out of the sand that rolled right up to tap-in range. He walked up and converted his par putt, giving him his first victory on the PGA Tour.
'I don't know what to say, don't know what to do,' said Short, who is playing this year on a major medical exemption. 'I'm so excited. Can't wait to go get a beer.'
Furyk squandered a chance at his fourth victory in Las Vegas and ran his playoff record to a disappointing 1-6.
'I'm obviously disappointed,' said Furyk, who won this title in 1995, 1998 and 1999. 'It was my tournament to win again. I didn't get it done on the last hole.'
Short shot a final-round, 6-under 66 on Sunday to join Furyk at 21 under par. Furyk fired a 7-under 65.
Overnight leader Ted Purdy was in the hunt until his tee ball at the 17th hit a sprinkler head and fell into the water. He finished with a 1-under 71 and tied for third place with Harrison Frazar, who posted a 69 on Sunday. The pair came in at 19-under-par 268.
Charles Howell III birdied the last en route to a 2-under 70. He took fifth place at minus-18.
Short moved up the leaderboard with birdies at the third, sixth and seventh holes. He dropped a shot at eight, then atoned for the miscue with a birdie at the par-5 13th.
Furyk took the lead with a chip-in eagle at the 16th, but Short followed him with another spectacular shot. Short's second stopped 8 feet past the flag, and the 41-year-old rolled in the eagle putt to get within two of Furyk.
Then came the two-shot swing at the 72nd hole.
Short, who got into the field this week because Arron Oberholser withdrew, had two major goals coming into the tournament -- to finish in the top-10 to get into the field next week and to earn his PGA Tour card.
'I just don't like to go to Tour School,' said Short. 'I've been there so many times. It's so hard. That was my main goal. I thought if I finished third, I could probably not have to go.'
Nick Watney (66) and Shigeki Maruyama (68) tied for sixth place at 17-under-par 270.
First and second-round leader Briny Baird rebounded from a third-round 78 with a 65 on Sunday. He shared eighth place with Will MacKenzie (67), Hidemichi Tanaka (69) and Steve Lowery (72). The group came in at minus-16.