Wiebe and Langer, who suffered a double bogey on the final hole of regulation, tied after 72 holes at 9-under-par 271 and had pars on the first two sudden-death playoff holes Sunday before play was called because of darkness.
They returned to the par-4, 473-yard 18th hole at Royal Birkdale early Monday and had pars and then bogeys to force a fifth playoff hole.
Wiebe, who found the rough with his drive, came close on a long birdie try and tapped in for par. Langer was in a greenside bunker in two and left himself about 10 feet for par to extend the playoff but missed the putt.
"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," Wiebe said afterward.
It was the fourth career victory on the Champions Tour for Wiebe, 55, and his first since the 2011 Greater Hickory Classic at Rock Barn. He'd won twice in his PGA Tour career.
Langer had been seeking his third major title on the Champions Tour. His wins came in 2010 at the U.S. Senior Open and the British Senior Open. Monday marked the fourth time he was the runner-up in the senior major.
Wiebe had a fourth-round of 4-under 66 but it was Langer's 6 on the par-4 18th that forced the playoff. He was 3-over in his last five holes of regulation Sunday.
Corey Pavin, with a final round of 65, and Peter Senior, with a 66, tied for third.