Tomas Tatar scored in the sixth round against Braden Holtby for Montreal (14-8-9), which has a point in four straight games (2-0-2).
"It was a big roller-coaster game, so we stayed with it and it's a big two points for sure," Tatar said.
Holtby made 36 saves in his first start in nine games for Vancouver (16-16-3), which was 7-1-0 in eight starts by Thatcher Demko since Holtby's previous start (March 2, a 5-2 loss at the Winnipeg Jets).
"It was a tough game, give Montreal credit, they played well tonight," Canucks coach Travis Green said. "It was a hard-fought game. I like that we battled back. We weren't playing well, I didn't like the second half of the second period, and a hard game for our team and a point well earned."
Carey Price, who backed up Jake Allen in a 3-2 overtime loss to Vancouver on Friday, made 14 saves and allowed one goal in six shootout attempts. He is 5-1-2 in his past eight starts.
The Canadiens were 0-9 after regulation this season (0-6 in overtime, 0-3 in shootouts).
"It's definitely nice to get the proverbial monkey off our back," Price said. "So let's move on, and if we have another one we'll try to win that one."
Brendan Gallagher scored for Montreal to tie it 4-4 at 5:29 of the third period with a wrist shot off a face-off win by Phillip Danault.