Montreal had 18 shots and made the most of very little offense against Henrik Lundqvist.
The Canadiens tied it late in the second period on Max Pacioretty's goal and went ahead for good when Galchenyuk matched Pacioretty with his second of the season 1:48 into the third.
Carey Price was only slightly busier than Lundqvist, and made 24 saves for his third win in the Canadiens' surge. Peter Budaj won the previous two, including a 3-0 home victory over Carolina on Monday.
Raphael Diaz added an empty-net goal with 1:06 remaining for the Canadiens, who have outscored opponents 15-5 in their streak.
Defenseman Anton Stralman netted his second of the season in the second period, but it wasn't enough for the Rangers, who played without top-line forward Rick Nash and had a five-game point streak snapped (4-0-1).
Nash was ruled out about an hour before the game because of an undisclosed injury that threatened to keep him out of New York's win Sunday against Washington.
Brian Boyle nearly tied it with about 6 minutes left, but his short-handed chance hit the post. New York received its second power play with 3:06 left but failed to take advantage.
Despite registering only 10 shots in the first two periods, Montreal entered the third locked in a tie.
Galchenyuk finished off a play that began with an odd-man rush. Rangers captain Ryan Callahan got back and had a chance to break it up, but couldn't. Lars Eller moved the puck down to former Rangers forward Brandon Prust near the right post, and Prust quickly got it in front to Galchenyuk, who lifted a rising shot that caromed in off the crossbar.
The Canadiens got even with 1:15 left in the second when Josh Gorges carried the puck in the Rangers end along the left-wing boards and flung a pass to the middle. Pacioretty wound up and ripped a drive from the high slot that beat Lundqvist inside the right post.
Pacioretty, who missed four games from Jan. 27-Feb. 2 because of appendicitis, scored for the second straight night. He has two goals and three assists in the past three games.
Stralman had given New York the lead with his second of the season, scored with 7:31 left in the second period. During a delayed penalty, the Rangers took advantage of an extra skater on the ice. Derek Stepan made a pass from the left side down to Stralman at the right post. The defenseman's first shot was stopped by Price's left pad, but Stralman then knocked the rebound past him.
The Canadiens didn't arrive in New York until the early morning hours Tuesday, following their win Monday.
Neither the Rangers nor Canadiens showed a whole lot of drive in an uneventful first period that featured no goals, one penalty and a total of eight shots -- including five by New York. Montreal didn't record its first shot of the game until only 3:49 remained in the first, when Prust pushed the puck in on Lundqvist from in-close.
Ryan McDonagh had the Rangers' final shot of the period with 5:20 left.
NOTES: Stralman scored his two goals this season in a four-game span. ... Montreal has had five shots in the first period in each of its past three games. ... Carl Hagelin earned an assist for New York, but his four-game goal streak was snapped. ... Montreal was whistled for two penalties. The Rangers were called for only one. ... Brandon Mashinter and Chris Kreider were in the lineup for Nash and Darroll Powe (concussion).