Both players found the putting surface with their third shots to the par-5 17th, the second extra hole. Weibring missed his birdie putt from just inside 10 feet. Jenkins sank his 5-foot birdie put for his sixth Champions Tour crown.
'I've been playing some good golf for the last month and a half,' said Jenkins, who has finished in the top-5 in five of his last six starts. 'I knew I was coming close to winning. My goal now is to win twice in a year, I've never done that.'
Mike Reid, who held the second-round lead and won the Senior PGA Championship last week, could only manage an even-par 71 in the final round. He finished alone in third at 7-under-par 206. Bob Gilder, Dave Barr, Mike McCullough and Dana Quigley shared fourth place at minus-6.
Jenkins two-putted for birdie on the par-5 first to get to 6 under. However, he stumbled to a bogey at the sixth. The Texan dropped in a 12-foot birdie putt on 11. He came back with an up-and-down birdie on the 13th to get to 7 under.
The 57-year-old moved to minus-8 with a birdie on the 15th. Jenkins joined Weibring at 9 under with a birdie on 16. Jenkins parred his final two holes to remain there.
'It's been a fabulous week,' said Jenkins. 'Everyone wasn't quite sure about the course when they came here. It's a very playable course. It turned out to be a great test of golf.'
Weibring, who won the Bruno's Memorial Classic earlier this year, also opened with a birdie on the first. He moved to 6 under with a 5-foot birdie putt on the fourth.
The 52-year-old ran home back-to-back birdie tries from the seventh to take the lead at minus-8. Weibring parred his next eight holes before an up-and-down birdie on 17. He parred the last to head to extra holes.
'It was very exciting with the wind,' Weibring said. 'I felt good about the way I played. I didn't make a bogey the last two days, including the playoff.'
Bruce Fleisher, Raymond Floyd, Mark James, Rodger Davis, James Mason and Pete Oakley shared eighth place at 5-under-par 208. That was Oakley's first top-10 finish since his win at the Senior British Open last summer.
Ed Dougherty, Mark Lye and Mike Sullivan were one stroke further back at minus-4.