Henrik Lundqvist, who made his 60th start of the season, stopped Andrei Kostitsyn on Montreal's first attempt after making 26 traditional saves.
New York moved one point ahead of Montreal in a tightly contested race for the final playoff spots in the Eastern Conference.
Nik Antropov, Nikolai Zherdev and Ryan Callahan all scored to stake the Rangers to the lead.
Andrei Markov's goal with 4:53 remaining in the third drew Montreal even at 3.
Markov, who assisted on Alex Kovalev's power-play goal late in the second, beat Lundqvist with a sharp-angled shot from deep in the left circle.
Maxim Lapierre also scored for Montreal, but Montreal fell to 1-1-2 under general manager Bob Gainey, who moved behind the bench after firing coach Guy Carbonneau on March 9.
The Rangers' NHL top-ranked penalty-killing unit was tested by a tripping call against Scott Gomez with 1:44 left in regulation.
Antropov scored for the third straight game 7:13 into the second to put New York up 1-0. The Russian forward beat Price during 4-on-4 play for his fourth goal in seven games since the Rangers acquired him from Toronto.
Kovalev, who returned after missing two games because of the flu, got Montreal even at 1 during a power play late in the second.
The Canadiens took advantage of Lauri Korpikoski's delay of game penalty for putting the puck over the glass.
Kovalev, who had two stints with the Rangers, scored his 17th goal at 18:26. He snapped a one-timer from the right side of the net to put Markov's cross-ice pass from the left point past Lundqvist and inside the right post.
Zherdev gave New York its second one-goal lead at 5:32 when he took Gomez's drop pass and put a wrist shot past Price from the right side that appeared to handcuff the goalie.
Lapierre tied it just 40 seconds later when he beat Lundqvist with a shot from the slot at 6:12.
Callahan extended his point streak to four games when he put a backhander through traffic and past Price 10:58 into the third.
Guillaume Latendresse assisted on Markov's goal in his return after missing 19 games because of a shoulder injury.
Notes: Lundqvist made 50 starts as a rookie in 2005-06, 69 in 2006-07, and 72 last season. ... Callahan has two goals and three assists during his streak. ... Rangers LW Sean Avery ended a three-game goal streak. Avery had four goals during his streak, and Callahan had a goal and three assists. ... The Canadiens had scored six goals in each of their previous three games against the Rangers. ... Montreal LW Alex Tanguay missed the game due to the flu.
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