Langer bogeyed his fifth attempt at the 18th hole on Royal Birkdale while Wiebe made par to claim the trophy. Storms delayed Sunday's fourth round and play was halted after two playoff holes because of darkness.
"I'm a little tired, I'll tell you that," Wiebe said. "I try to play the playoffs like I was playing the tournament. I just got a few extra holes in. .... I have never been in a playoff in a major championship either so I was a first-timer. I just tried to play the golf course as good as I could."
Langer, the two-time Masters champion, had an immediate chance to secure his second Senior British Open victory when play resumed at 8 a.m. (3 a.m. ET) but missed a 12-foot birdie putt.
Two holes later, Wiebe made a superb approach shot to set up a two-putt for par, while Langer failed to get up-and-down.
"I was surprised that Bernie missed his putt (on the third playoff hole) because he's a great putter, he's a great player," Wiebe said. "It just was my time."
Langer -- who won the tournament in 2010 -- held a three-shot lead going into the fourth round and was two strokes ahead with one hole to play, but finished with a double bogey. Wiebe, meanwhile, shot a 4-under 66 to match the German veteran 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."
It was the first time in the 27-year history of the Senior British Open that the tournament finished on Monday.
Wiebe said it was strange to play in near darkness as the playoff started in failing light on Sunday.
"I have never done that so that was unusual but I think we would have both liked to have played because of our schedule," Wiebe said. "Obviously it was really dark at that second hole and mother nature always wins. I just figured that this is par for the course and we will go at it again at eight and here we are."