Kuchar closed with an even-par 72, while Claxton fired a 3-under 69 to end at 9-under-par 279.
'I didn't pay attention to where I stood, but boy this was exciting,' said Kuchar, who collected $81,000 for the win. 'It was great to be back in this position. It's been a while and it feels sweet again. It's bittersweet to beat a friend, but a win's a win. Paul played great.'
The pair returned to the par-5 18th for the first extra hole. Kuchar and Claxton both two-putted for birdie from over 65 feet out.
They moved to the tough par-3 ninth for the second playoff hole. Kuchar, who birdied the hole in regulation, dropped his tee ball 12 feet from the cup. Claxton's tee shot came to rest 2 feet closer to the hole.
Kuchar missed his birdie putt on the low side and tapped in for par. Claxton's birdie effort slid over the left edge and it was back to the 18th at The Dominion Club.
Both players found the right rough off the tee at the par-5. Claxton knocked his second just over the back edge of the green. Kuchar's approach just cleared the water fronting the green and came to rest within 12 feet of the hole.
With the pressure on, Claxton hit his chip thin and ran it through the green. He pitched his fourth to 2 feet.
Kuchar closed things out. He rolled his eagle effort just right of the cup, but calmly kicked in the birdie try for his first tour win. Kuchar also owns one victory on the PGA TOUR.
'It was a cut 7-iron off the bank and it rolled right around the cup. Just like I planned it,' joked Kuchar of his approach on the third extra hole.
Jason Caron closed with a 2-under 70 to end in third place at 8-under-par 280.
Third-round leader Jeff Burns struggled to a six-over 78 to share fourth at minus-6. He ended alongside Steve Wheatcroft (72), Charlie Wi (72), Erik Compton (76), Bradley Hughes (69) and Andrew Pratt (72).
Kuchar birdied the second to get to 10 under, but gave that stroke right back with a bogey on three. He dropped another stroke on eight.
The 27-year-old atoned for that error with a 5-foot birdie putt on the ninth to get back to minus-9. He shared the lead at that point with Peter Tomasulo.
Kuchar bogeyed the 17th after missing the green, but drained a 25-footer for birdie on 18 to join Claxton at minus-9.
Claxton moved to 8 under with birdies on two and six. He gave a stroke back to par with a bogey at the eighth. The 2001 Louisiana Open winner birdied the 13th to move to minus-8 and closed with a birdie on 18 to end at 9 under.
Tomasulo was at 9 under through eight holes after two bogeys and one birdie in that span. However, he carded a birdie, two bogeys and two double-bogeys the rest of the way to tumble down the leaderboard.
Burns, who was looking for his first tour win, posted three bogeys on each nine to fall to minus-5.
Tomasulo, Steve Marino and Fran Quinn shared 10th at 4-under-par 284.