Sutter was trailing behind as James Neal and Simon Despres spearheaded a rush into the Montreal zone on the decisive play early in the extra frame.
He wristed Despres' drop pass toward the net and beat Carey Price with the game-winner -- his second goal of the game -- enabling the Penguins to break a two-game losing streak.
Each team held a one-goal lead in the third period after the Canadiens had battled back from a 4-2 deficit in the middle frame.
Tomas Vokoun got the win despite yielding six goals on 39 shots.
Brian Gionta tallied twice while Max Pacioretty added a goal and an assist for the Canadiens, who fell for the second time in three contests.
Price ended with 29 saves on 36 shots in the loss.