Grigorenko beat Zachary Fucale with a backhand on Buffalo's second attempt in the tiebreaker.
The Sabres, who never led, erased four one-goal deficits. Colton Gillies got the Sabres even at 4 with 4:24 left in regulation.
Zemgus Girgensons scored two goals in the second for Buffalo, making it 3-3 at 11:51 on the first shot faced by Fucale. Jamie McBain also scored for the Sabres.
Michael Bournival scored twice for the Canadiens. Martin St-Pierre netted Montreal's second power-play goal at 12:59 into the third to give the Canadiens a 4-3 lead.
Erik Nystrom added a power-play goal for Montreal.