Vanek had the decisive shot after Briere and Alex Kovalev traded goals, beating Cristobal Huet with a slapshot from the slot. Sergei Samsonov missed on Montreal's last attempt.
Michael Ryder and Sheldon Souray scored for Montreal (16-8-5), while Jiri Novotny and Derek Roy replied in regulation for Northeast Division-leading Buffalo (22-5-1), which was coming off a 3-1 loss in Florida on Thursday.
With Montreal trailing 2-1 going into the third period, Souray tied it with a power-play goal at 2:52. He moved in from the point, faked a shot, then beat Ryan Miller with a high shot.
It was Souray's team-leading 11th goal of the season, but the defenceman left 11 minutes into the third with an apparent wrist or arm injury after being checked into the boards by Paul Gaustad.
The Sabres lacked their usual zip in a scoreless opening period in which they were outshot 11-5 and leaned on Miller to keep from falling behind.
Miller was sharp again to stop Saku Koivu in alone shorthanded to open the second and the Sabres got back on track.
Novotny went to the net to put in Maxim Afinogenov's shot to open the scoring with the teams each down a man at 6:12.
Montreal tied it on a power play at 9:11 when Ryder slid in the rebound of a hard shot from the point by Souray.
But with 2:15 left in the period, a run of pressure in the Montreal zone ended when Roy lifted a shot inside the near post.
Buffalo's Daniel Briere was called for diving in the second after an emphatic fall from a push by Craig Rivet. It was Briere's first diving call and he will be warned by the league. A second offence carries a US$1,000 fine.
NOTES: Defenceman Teppo Numminen broke a toe when he blocked a shot Thursday in Florida and is listed as day-to-day. Buffalo recalled Andrej Sekera, who saw his first NHL action for the Sabres, and returned Dan Paille to Rochester of the AHL. Henrik Tallinder remains sidelined with a broken arm . . . Janne Niinimaa, who took a puck in the eye on Long Island, N.Y., on Thursday, sat out, while Garth Murray was a healthy scratch for Montreal . . . F Chris Higgins, who has missed 16 games with a sprained ankle, skated with the Canadiens on Saturday morning and is expected to return next week.