BUFFALO, N.Y. -- The Buffalo Sabres cut it about as close as possible.
Miller made 26 saves to help the Sabres (40-22-10) win their fourth straight game.
"It was an ugly game but we'll take the two points," Vanek said. "We talked about getting pucks to the net and going hard to the net and capitalizing on some rebounds. It took 58 minutes to finally get one. I think we all know that we can be better, and we will be better."
Up until that 58-minute mark, the Sabres were being thwarted by Canadiens goalie Carey Price, who finished with 40 saves in his first start since being benched after allowing three first-period goals against the Anaheim Ducks seven games ago. Price was steady throughout but was undone when Buffalo blitzed him with six skaters for the majority of the final 3½ minutes of regulation.
Connolly scored with Buffalo enjoying a six-on-four advantage with Miller out for an extra attacker, and Montador scored his fourth of the season in a scramble in front of Price to tie it at 2.
"I'm not happy whatsoever," Price said. "Not one bit. We got to find a way to get that done. That's a point we need, and we need to find a way to do it. We played hard for 55 minutes and after that we just sat ... and let them come to us."
After Pominville scored in the first round of the shootout, Vanek iced it in the third round with a wicked snap shot that beat Price. Miller gave Buffalo the bonus point by stopping Montreal's Andrei Kostitsyn and Mike Cammalleri in the first two rounds.
"We got that one on the power play, and that was our hope right there," Miller said. "Then we got one six-on-five and then got the job done in the shootout."
Kostitsyn scored twice for the Canadiens (36-30-8), who've lost three straight after a six-game winning streak.
Montreal was stung by giving the Sabres six consecutive man-advantage opportunities starting with under seven minutes to go in the second period.
"Carey gave us a strong game, and you'd like to win that game, but the penalties made it difficult for us," Montreal coach Jacques Martin said. "What you've got to learn from that is the penalties cost us."
The Canadiens, 4-1-2 in their last seven on the road, received a boost with the return of leading scorer Cammalleri and defenseman Marc-Andre Bergeron from injuries. Bergeron assisted in Kostitsyn's second goal.
Kostitsyn popped in his second of the game on the power play 4:57 into the second period to stretch Montreal's lead to 2-0. From the left circle he wheeled around and wristed a shot that handcuffed Miller.
Kostitsyn scored just 41 seconds into the game to give Montreal a 1-0 lead. Wide open and standing to the right of Miller, Kostitsyn easily tapped in Tomas Plekanec's pass from the left of the net.
The Sabres won their 40th game of the season, the fourth time they've reached the plateau in the last five seasons. ... Kostitsyn snapped a 13-game goal drought. ... The Sabres opened a three-games-in-four-nights homestand. ... Cammalleri had missed the past 17 games because of a knee injury, while Bergeron was forced to miss 15 games with a knee injury. ... Montreal left winger Travis Moen didn't play after he was cut on the face by a skate in the Canadiens loss to the Ottawa Senators on Monday night. ... Buffalo left winger Raffi Torres missed his third straight game because of an undisclosed injury. ... Connolly missed the last half of the first period after being hit in the face by Kostitsyn's clearing attempt. ... Sabres right winger Patrick Kaleta missed the final period with neck soreness.