Sindelar, 46, birdied two of the last three holes, waited for Oberholser to wilt, then polished him off with a par on the second extra hole for his first victory since the 1990 Hardee's Golf Classic.
'I never, ever gave up hope,' Sindelar said. 'This feels way too good. It's a thrill. It will take me a while to understand this is real.'
Oberholser might need some time to recover from a stunning collapse at Quail Hollow.
Two shots ahead with three to play, his first PGA Tour victory firmly in his grasp, the pressure overcame him. He bogeyed the next two holes and needed a brilliant recovery from the trees on No. 18 just to force the playoff.
Sindelar closed with a 69, while Oberholser had a 72. They finished at 11-under 277.
Tiger Woods, errant as ever, gave himself a chance with a 30-foot birdie putt on the 17th and nearly chipped in for birdie on the 18th. He closed with a 68 and joined Carlos Franco one shot behind. Franco had a 35-foot birdie putt on the final hole that caught the right lip. He shot 70.
Phil Mickelson had a bogey-free 67, the best round of the day, to finish in a tie for fifth.
Vijay Singh held out hope for a third consecutive victory and got within one shot of the lead with an eagle on the 15th hole, but a bogey-double bogey finished dropped him into a tie for 10th, four shots behind.
Oberholser appeared to have this locked up. He never lost his lead all day, and when his 5-wood into the 15th hole stopped 8 feet away for eagle, he pumped his fist with passion.
But those finishing holes can expose the slightest nerves, and Oberholser was loaded with them.
He hooked his tee shot on the 16th next to a bush, had to punch out to the fairway and then overshot the green. Only a great chip allowed him to escape with bogey.
Then, he badly pulled his tee shot on the 17th and got a huge break - the ball was one turn away from going in the water. From the hazard line, he chipped to 10 feet and missed the putt.
Suddenly, he was tied at 11 under with Sindelar, who two groups ahead made a terrific charge. He holed a 15-foot birdie on No. 16, then hit his approach to 3 feet on the par-3 17th, the hardest hole at Quail Hollow.
A playoff looked out of the question when Oberholser, protecting from the stream down the left side, blocked his tee shot into the trees. With a large pine causing him to alter his backswing, he hit a high shot that squeezed through a gap in the trees and stopped 45 feet from the cup for a two-putt par.
Both players made par on No. 18, and they went to No. 16, where Oberholser's troubles started. The second time around was no different. He pulled his approach into the bunker, blasted out to 10 feet and had to make a 5-footer for bogey. Sindelar ran his 30-foot birdie putt about 4 feet past the hole, but calmly rolled it in.
It was his seventh career victory and paid off $1.08 million, more than he had earned in any of his 20 previous seasons on the PGA Tour.
Sindelar's victory drought was the third-longest on tour behind Butch Baird and Ed Fiori.
Oberholser played in the final group for the second time this year. He was tied with Singh at Pebble Beach and struggled to a 76. This time, he vowed to play the course, and not any other player.
He turned out to be his own worst enemy.
'I'm really not that disappointed,' Oberholser said. 'Under the pressure, under the heat I played in today, I think I did a great job.'
Mickelson now has finished in the top 10 in all but one out of the 11 tournaments he played, and he has been around the lead every Sunday he plays. This was no exception. Starting the final round seven shots behind, he played bogey-free and quietly crept into contention.
'It just looks like it's going to fall a little shy,' Mickelson said. 'I just struggled on the greens, just didn't read them well. Other than that, it's been a great week.'
Woods holed a 30-foot putt on the par-3 17th to get to 10 under. He took a few steps back and jabbed his fist when it dropped, knowing he had at least had a chance.
Moments later, Singh chipped in from 30 feet short of the 15th green for eagle, putting him at 10 under. It was an eagle on the 15th hole at English Turn last week in New Orleans that sent Singh to victory, and the way he pumped his fist showed that he expected more good things to follow.
Two holes later, however, Singh missed the green to the right on the 17th and took bogey, then chopped up the 18th hole for a double bogey that dropped him into a tie for 10th.
Woods will look back at the 14th and 15th hole, the best two birdie chances on the back nine. He parred them both.
'I didn't play the par 5s well at all today,' Woods said. 'But I hung in there, made a couple of putts when I needed to. All I wanted to do was give myself a chance, and at least I did that.'
Divots:@ Davis Love III was all over the map Sunday, but wound up where he started - an even-par 72. Love made two triple bogeys on the front nine for a 40, then had a birdie-eagle-birdie stretch on the back nine for a 32. ... The double bogey by Singh on the 18th hole enabled Lucas Glover, who had been playing on a sponsor's exemption, to finish in the top 10 and get into the Byron Nelson Classic next week.
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