Galchenyuk, who played his third straight game at center, tied the game 1-1 when he scored at 9:09 of the second period, nine seconds after Nicolas Deslauriers gave Buffalo a 1-0 lead.
He scored his second of the game, his 11th in the past eight games and team-leading 25th, on a power play with 44 seconds remaining in the period.
Galchenyuk, who had a chance at a hat trick but missed his shot at an empty net, is the first Canadiens player with three consecutive multiple-goal games since Brian Savage from Oct. 25-28, 1995.
"We knew he came into the game red-hot, we knew he was on fire and unfortunately we were unable to stop him there," Bylsma said. "He had the power-play opportunity there; that one, disappointingly enough, they had us on the ropes there for a lot of that power play and we blocked a shot there and it looks like we might get a reprieve, and kind of a bang-bang play. The puck bounces off of us and then it's quick out to the front of the net, and Alex, he's all alone, all by himself standing at our goalie. For a guy that's that hot, it's a tap-in for him."
Brian Gionta got the Sabres even at 2-2 with his ninth goal at 5:10 of the third period. Evander Kane's shot from the left side hit the skates of the Sabres' captain and deflected into the net.
But Mitchell scored his eighth goal at 13:21 to make it 3-2. He snapped a shot past Robin Lehner from the lower right circle.