Bo Horvat, Alexander Edler, and Jake Virtanen also scored for Vancouver.
Toffoli has three goals and five points in his first three games since the Canucks acquired him from the Los Angeles Kings over a week ago. Toffoli also has seven goals in his last five games, including his final two games in Los Angeles.
''It's been incredible,'' Toffoli said. ''The guys have been great. Getting points in all three games has made it a lot more fun.''
Paul Byron, Shea Weber, and Jordan Weal scored for Montreal.
Thatcher Demko made 37 saves for his 11th victory of the season. Carey Price made 29 saves in the loss.
''I want the guys to have confidence in me,'' Demko said.
Demko, along with newcomer Louis Domingue, will be relied on in goal for the Canucks next month after they lost goaltender Jacob Markstrom to a knee injury on Saturday against Boston.
''We don't look at it like it should change the way we play,'' Canucks forward J.T. Miller said. ''We know what makes our team good. We try to stay focused and not be distracted. Obviously, we'd like to have (Markstrom) in net. But we know we have a ton of confidence in whoever's back there. And even picking up (Domingue), we're going to have confidence if he gets a game.''
Montreal jumped out to a 2-0 lead, scoring both in a 90-second span early in the first period. Byron scored first, his third of the season, after Max Domi sped up the wing into the Canucks' zone and sent the puck through the crease and onto Byron's stick.
Weber added to the lead with his 15th of the season. The Canucks called a timeout after allowing two goals on six shots, giving head coach Travis Green a chance to push a strongly worded message to his team.
''Some swear words mixed in there,'' Green said. ''We had to get a little better with the puck, a little stronger on the puck. A little more direct in our game. We dug ourselves in a hole early. It was nice to get two points tonight.''
Vancouver scored with 4:03 left in the first, a power-play goal from Horvat. The Vancouver captain has scored at least 20 or more goals in his last four seasons.
Edler tied the game at 2-all in the second period with a shot from the blue line.
Weal gave Montreal the lead once more with his seventh of the season, 50 seconds into the third period. Domi picked up his second assist.
Virtanen tied the game at 5:53 of the third.
NOTES: The Canadiens celebrated LGBTQ+ night at Bell Centre by wearing rainbow-colored jerseys and stick tape during the pregame warmup.
Canucks: At Ottawa on Thursday.
Canadiens: Host the New York Rangers on Thursday.
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