They will finish the playoff Monday morning.
Wiebe shot a 4-under 66 to match Langer at 9-under 271. Langer had a 70. They each parred the par-4 18th twice in the playoff.
"Playing in the dark was a new one for me," Wiebe said. "Hopefully, we'll figure it out tomorrow."
Langer, the 2010 winner at Carnoustie, blew a two-stroke lead with a double bogey on the final hole of regulation after hitting into a greenside bunker. Minutes earlier, Wiebe's birdie putt at No. 18 came up short.
"I gave it away," Langer said. "I could bogey the last hole and win. I thought I hit a decent shot in the bunker, but it came out too low.
"But I made two good pars in the playoff in the dark, so what can you say. I'm sorry to keep you all hanging round tomorrow."
Corey Pavin, Peter Senior and David Frost tied for third at 6 under. Pavin shot a 65, Senior had a 66, and Frost a 70.
Langer seemed in control most of the afternoon, making the turn in 2-under 32 and picking up another birdie at the 12th. But he dropped a shot at the 14th, before needing two shots to get out of the bunker and two putts on the green at the last.
Wiebe made five birdies and one bogey in his round.
In the first hole of the playoff, Wiebe two-putted from just off the green. Langer did the same from 4 feet closer.
They were offered the chance by tournament referee Andy McFeeto to call it a day, but opted to try again. With the clubhouse lights trying to brighten the gloom, the players were driven back to the tee.
Wiebe put his second shot almost exactly on the same spot where he hit to on the first playoff hole, while Langer was 20 feet away on the green. Wiebe putted up to 6 feet, Langer to 3. They sank their putts, shook hands and made plans to return Monday.
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