Montreal won its fourth in a row, a victory that came after the Canadiens had squandered a three-goal lead.
Boone Jenner's second goal of the game lifted Columbus into a 3-3 tie 4:35 into the third period and overtime was looming when Plekanec took charge.
He collected a centering pass from Brian Gionta before moving in on the goal and firing a wrist shot that got the best of Sergei Bobrovsky to give Montreal the lead. Plekanec added an empty-net score with 16 seconds to play.
Montreal built its lead on goals from Rene Bourque, Michael Bournival and Andrei Markov. The goal by Markov came shorthanded 6:52 into the second period, but by the time the period was over Columbus had closed within one on scores from Jenner and Ryan Johnson.
Carey Price earned the win with 30 saves. Bobrovsky stopped 31 shots.