Montreal headed to the room with a 4-0 lead, outshooting Vancouver 11-9 in the opening frame.
The second was a little quieter on the scoresheet, but Conor Garland broke Montembeault's shutout with 3:02 left to play in the period.
The Canucks further cut into the Canadiens' lead 90 seconds later, when Ilya Mikheyev beat Montembeault to make it a 4-2 game.
That's where things would stand after 40 minutes of play. Vancouver outshot Montreal 13-8 in the period.
In the final frame, Bo Horvat brought his team within one when he netted his 20th of the season at the 4:02 mark.
Then, Mikheyev evened the score with his second of the night just 1:43 later.
The Canucks took their first lead of the night with 11:11 remaining in regulation when Jack Studnicka lit the lamp for the second time this season.
The Habs weren't ready to roll over, however. First, Christian Dvorak tied things up again when he picked up Evgenii Dadonov's rebound and lifted it over Collin Delia - who replaced Martin after the Habs' fourth goal - to tie the game with 4:37 remaining.