The Canadiens scored four times in 9:53 to take a 5-4 lead before Auston Matthews tied it 5-5 with 1:15 remaining in the third period and goalie Michael Hutchinson pulled for the extra skater.
"That's what we're all about. We're resilient, we're going to work and continue to push," Canadiens forward Brendan Gallagher said. "We've been through this many times. We remember being in this building being up three goals (last season) and them coming back on us, so it's kind of nice to return the favor."
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Gallagher had a goal and two assists, Max Domi and Phillip Danault each had a goal and an assist, and Carey Price made 32 saves for the Canadiens (1-0-1). Paul Byron scored the only goal in the shootout.
"It wasn't a great start for us," Canadiens captain Shea Weber said. "For whatever reason, they were just on top of us for a couple periods. We didn't panic and just kind of hit the reset button."
Matthews scored two goals, Alex Kerfoot had a goal and two assists, Trevor Moore had a goal and an assist, and Hutchinson made 37 saves for the Maple Leafs (2-0-1).
"We were back to back (defeated the Columbus Blue Jackets on Friday), it's no secret probably we're going to be fatigued, but we can't let that get in the way of us making boneheaded mistakes and costing us opportunities on the other end that they're capitalizing on," Matthews said. "It's definitely something we can learn on. We've run into it in the past, but it's on us to make sure no matter how tired we are, we're making good decisions and taking care of the puck."