Chris Kreider streaked down the left wing and flung a pass through the slot to Zibanejad, who ripped a one-timer past Carey Price for the win.
The Rangers now lead the series 3-2 with a chance to clinch on Saturday night in New York.
Price stopped 33 shots for the Canadiens. Henrik Lundqvist made 34 saves for New York.
Artturi Lehkonen and Brendan Gallagher scored in regulation for Montreal. Jesper Fast and Brady Skjei replied for the Rangers.
Lehkonen's second of the playoffs opened the scoring at 12:07 of the first period. After Nathan Beaulieu's point shot hit the rookie and Lundqvist saved Lehkonen's first shot attempt, the young Finn outmuscled Marc Staal behind the net and tucked the rebound in on a wraparound.
Fast evened it up with a short-handed tally at 15:56. Zibanejad cut into the middle before feeding Fast, who beat Price between the legs.
But Gallagher got the crowd roaring again less than 30 seconds later on the same power play. Andrei Markov sent a perfect backhand pass to the winger in the high slot and Gallagher had plenty of time before firing one past Lundqvist.
The Canadiens, who struggled in the second period in this series, were on their game for the first half of the middle frame, helped by a couple of back-to-back power plays early on. Montreal held an 8-3 edge in shots on goal in the period before the midpoint but couldn't get anything by Lundqvist.
New York picked it up in the second half of the period and was rewarded at 18:28. Rick Nash curled in from the corner and Skjei picked up the rebound and shot through Shea Weber's legs into an open side of the net.
The Rangers picked up where they left off to start the third, getting the first four shots while Montreal didn't get its first until just past the halfway mark of the period.
The Canadiens had chances to take the lead when Phillip Danault hit the crossbar short-handed and Max Pacioretty was denied by Lundqvist on a breakaway.
NOTES: Montreal D Alexei Emelin suited up for the first time since suffering an undisclosed injury on April 5. ... The Canadiens scratched C/W Brian Flynn, RW Michael McCarron, LW Andreas Martinsen, D Brandon Davidson and D Nikita Nesterov. ... The Rangers scratched LW Matt Puempel, LW Tanner Glass, C Brandon Pirri, D Adam Clendening, D Kevin Klein and D Steven Kampfer.