Each golfer had registered a pair of pars over the first two extra holes before play was called at Royal Birkdale Golf Club following several weather delays.
Tournament officials said the playoff would resume at 3 a.m. (EDT) Monday.
Langer looked to have the Champions Tour major wrapped up heading into the final hole with a two-shot lead. But he stumbled to a double-bogey at No. 18, finishing regulation at 9-under-par 271.
The gaffe forced a playoff with Wiebe, who registered a 4-under 66 to match Langer atop the leaderboard.