Gaustad tipped Brad Boyes' centering pass past Carey Price 6:17 into the final frame, breaking a 1-1 tie.
Patrick Kaleta sealed the win on a tally with 52 seconds remaining and the Sabres took their second consecutive road win after dropping 12 straight away from home.
Price absorbed the loss despite stopping 37-of-39 shots. Max Pacioretty scored the lone goal for the Canadiens.
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