Carl Soderberg and Alexander Kerfoot scored to help Colorado match the longest home winning streak in franchise history, set in 1994-95 when the team was the Quebec Nordiques.
Antti Niemi stopped 23 shots for Montreal. The Canadiens are 8-17-1 on the road.
Varlamov's last win came against Toronto on Dec. 29, and he left his next start against Winnipeg with a groin injury. He missed 10 games before returning against the Jets on Feb. 3.
Wednesday was his fourth since coming back and one of his best of the season. He denied the Canadiens on several chances in the third for his second shutout of the season.
With Montreal pressing for the equalizer late in the third, Kerfoot scored at 16:58 to give the Avalanche a two-goal cushion.
Colorado had more than two consecutive minutes of play in Montreal's end early in the second period but didn't score until Soderberg beat Niemi with a wraparound to make it 1-0 at 15:55.
Game notes There was a moment of silence before the national anthems for the victims of the school shooting in Florida. ... The Avalanche recalled Rocco Grimaldi from San Antonio of the AHL. Nathan MacKinnon, out since Jan. 30 with a left shoulder injury, was placed on injured reserve to make room for Grimaldi on the roster. ... Montreal C Phillip Danault (concussion) is on the four-game trip but has not played since Jan. 13. ... This was the 3,000th game in Quebec/Colorado franchise history, with 1,744 having been played since the team became the Avalanche in 1995. ... Colorado LW Blake Comeau missed his third straight game with a lower-body injury.