'It just kind of came out of nowhere,' said Triplett, who collected $540,000 for the title. 'I haven't been playing great. I haven't been getting as much out of my rounds as I could.'
Kelly had a putt on the 72nd hole to tie, but missed. He posted a 7-under 65 to finish alone in second place at minus-21.
Third-round leader Duffy Waldorf held the lead on the front nine, but a poor back nine cost him the title. He only made one birdie, at the par-5 10th, then bogeyed 16 to lose any chance at victory.
He managed an even-par 72 to tie for third place with 2004 champion Heath Slocum (68) and long-hitting Bubba Watson (70). The trio came in at 19-under- par 269.
Watson became the first player to go 72 holes without a bogey since Lee Trevino at the 1974 Greater New Orleans Open.
Triplett flew out of the gate on Sunday with five consecutive birdies. He bogeyed seven, but got the stroke back with a birdie at the very next hole.
Triplett went on another birdie tear, this time going for four in a row from the 14th. After he birdied No. 16, he was ahead of the field, but Kelly charged hard up the leaderboard.
Kelly meanwhile tallied four birdies on the front nine, then rolled in a 19-foot birdie putt at the 12th. That birdie put him one behind Waldorf, but when Kelly drained a 25-footer for birdie at 13, he was tied for first.
Triplett surged ahead with his birdie run on the back nine, but Kelly kept pace. He kicked in a short birdie putt at the 16th, but ran into some bad fortune at 17 and it had nothing to do with him.
Triplett chipped in for birdie from close to 50 feet at the 17th. That moved Triplett to 22-under par and he had to wait to see if Kelly could catch him and force a playoff.
Kelly parred 17, but hit his approach to 11 feet at the last. His birdie effort moved right by the hole, meaning Triplett would visit the winner's circle.
'I got nervous,' admitted Triplett.
'There's always positives when you shoot 65 and finish second,' said Kelly. 'It's just the positives hurt a little bit more. I had all of my chances.'
David Branshaw carded a 4-under 68 on Sunday to finish alone in sixth place at 18-under-par 270.
Jason Gore and Doug Barron each shot rounds of 4-under 68 in the final round and tied for seventh place at minus-17. Cameron Beckman (66), Charley Hoffman (66), Bo Van Pelt (67), Gabriel Hjertstedt (70) and Mark Wilson (72) shared ninth place at 16-under-par 272.