The Ducks scored three first-period goals but the Canadiens came back with a goal in the second and two more in the third to force overtime.
Anaheim was up 1-0 going into the final round of the shootout when Brian Gionta put a wrist shot past Ducks' goalie Jonas Hiller. James Wisniewski was unable to score for Anaheim and Plekanec beat Hiller for the game-winner.
Plekanec also scored a second-period goal for Montreal, which also got regulation goals from Gionta and Andrei Markov.
Jaroslav Halak -- who replaced Carey Price after Price allowed three first-period goals -- finished with 21 saves for the Canadiens.
Corey Perry, Lubomir Visnovsky and Scott Niedermayer scored for Anaheim, which has lost three straight since league play resumed following the Olympic break.
Hiller finished with 40 saves for the Ducks.
2014: The Year in Music [PHOTOS]