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.
Beautician charged with giving client fatal silicone butt injection
Duggar sisters unveil Christian dating rules in new book