Wiebe took advantage of Langer's failure to close out the tournament, beating the German on the fifth playoff hole Monday at Royal Birkdale for his first senior major title.
The American used a superb approach shot from the rough to set up a two-putt for par, while Langer failed to get up and down, seeing his par putt stay out.
"I'm speechless," Wiebe said. "I think it's always better for both players had there been a birdie to win the playoff instead of a bogey, but right now, I don't really care. I'm glad it's over, and I'm honored.
Langer led by two shots going into the final hole of regulation Sunday, only to settle for a double bogey when he struggled to get out of a bunker. The playoff was then halted after two holes because of darkness and resumed Monday, with Langer immediately missing another chance to win when his 12-foot putt wouldn't drop.
"I just was luckier today and last night than Bernie I guess," Wiebe said. "I also feel like Bernie has won, what, a couple hundred tournaments. He's won so many, I feel like this was my turn."
Langer, who won the tournament in 2010, was up by three shots entering the fourth round. He looked certain to earn a second Senior British Open title when he teed off at the 72nd.
But his approach shot landed in a greenside bunker, and he needed two strokes just to get out of the sand. Wiebe, meanwhile, shot a 4-under 66 to match the German at 9-under 271.
"Obviously, not what I wanted. But the major mistake was again yesterday, the 72nd hole," Langer said. "In the playoff anybody can win. It comes down to one good shot or one bad shot. And that's what happened. Mark is a very deserving champion."
This was the first time in the 27-year history of the Senior British Open the tournament finished on a Monday.
Former U.S. Ryder Cup captain Corey Pavin shared third — three shots back — with David Frost of South Africa and Peter Senior of Australia. It was the third year in a row Pavin tied for third at the tournament.
Wiebe's season has been plagued by a back and arm injury and he hasn't finished in the top 10 on the U.S. Champions Tour all year. But he said he could swing a club without pain again, which "helped me so much with my attitude."
"Once I realized I could play and it didn't hurt very much to swing," Wiebe said, "I was instantly in a great mood, and I knew I was playing good."
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