Theodore, who got a rare night off Friday, made 29 saves, including one with 14.8 seconds remaining that led to a pileup of players from both teams as the Canadiens' goalie kept the puck from crossing the line.
"I didn't see much," Theodore said. "I know I stopped it but I didn't know where it was, and then all of a sudden I saw two guys coming at me hard but a good thing it was Danny -- he weighs, what 150? 160? -- so it wasn't that bad."
Buffalo has lost three straight, and five of six, including a 10-4 drubbing at home against Ottawa on Wednesday night.
"It's just frustration," Briere said. "We don't know where to go anymore. We don't know what is a penalty, and what's not. We're trying to bang some rebounds in, just going to the net. I paid the price for trying to score a little too hard."
Montreal improved to 5-3-0 at the Bell Centre, and 11-3-1 overall. The Canadiens' 6-0-1 road mark following Friday's 3-2 win in Buffalo is the franchise's best road start since it began play in 1909, eight years prior to the NHL's formation.
"These points are huge," Koivu said. "We were fortunate enough to get the win even though we didn't play well last night in Buffalo, but I think it's a very positive thing that we can still get some points even though we were not playing our best hockey tonight."
"We didn't play the game we wanted to play tonight, and that cost us another two points," Dumont said. "We have to make sure that we come back even harder on Wednesday at home against Carolina. That's going to be another big two points, but it's really disappointing to lose four points like that in games that we should have won."
Canadiens forward Jan Bulis left the game with a shoulder injury in the first. He did not return.
Kovalev scored his fifth of the season 5:03 in to put Montreal up 1-0. After Dumont scored on a power-play late in the first, Kovalev restored the Canadiens' lead 7:22 into the second on a power play, beating Martin Biron stick side with a wrist shot for his second of the game.
Zednik made it 3-1 with another power-play effort early in the third, his third goal in five games since returning from a groin injury.
Afinogenov got his second assist and helped draw the Sabres within one when Gaustad redirected a shot past Theodore at 11:21.
Koivu was called for roughing as the Canadiens drew an extra penalty on the late pileup.
- Sabres RW Mike Grier returned to the lineup after missing Friday's game with a hip injury.
- D Jeff Jillson dressed for Buffalo after being recalled from Rochester of the AHL earlier in the day. RW Jason Pominville, who made his NHL debut Friday, was assigned to Rochester.
- The Canadiens have sold out each of their eight home games.
- Montreal C Radek Bonk missed his fourth straight game because of a groin injury.
- Koivu has a goal and six assists in his last four games.