Brendan Gallagher had a goal and two assists for Montreal, Phillip Danault and Max Domi each had a goal and an assist each. Jeff Petry scored on a penalty shot, Jonathan Drouin also scored, and Price made 32 saves.
"He's unbelievable," Domi said about Price. "He's the best goalie in the world for a reason."
Matthews scored twice for Toronto -- his fourth and fifth goals of the season -- and Alexander Kerfoot had a goal and two assists for his first three points with Toronto. Trevor Moore added a goal and an assist, William Nylander scored, and Michael Hutchinson stopped 37 shots.
After a wild third period that saw Toronto blow a 4-1 lead before tying it at 5, Price robbed Marner on a breakaway in overtime before the Canadiens came close at the other end on a scramble that resulted in a slashing penalty to Tavares that helped save a goal.
The Leafs managed to kill that one off and Tavares had a chance on another break coming out of the box, but Price snagged his effort with the glove. Domi then hit the crossbar on a 2-on-1 rush to cap a frantic extra session.
Toronto appeared to have the game in hand when Nylander made it 4-1 at 5:16 of the third period on a power play, but Montreal scored 11 seconds later when Drouin's pass in front went in off Leafs defenseman Morgan Rielly.
Gallagher made it 4-3 at 6:29 after an icing, banging home a rebound.
Kerfoot took an interference penalty with under nine minutes left in regulation that Toronto came close to killing off, but Leafs winger Kasperi Kapanen inexplicably threw his broken stick at Petry after blocking a shot when the Canadiens defenseman had the puck at the point.
Petry was awarded a penalty shot as a result, and he beat Hutchinson glove-side to tie it at 13:02.
"I've seen guys throw a stick at the puck, but never at somebody," Petry said. "No one knew the ruling on it. I saw the ref's arm go up. They originally said anybody can take it, and then they changed it and said anybody on the ice. And then they said it had to be me, so it was a weird way it played out."
Kapanen said fatigue played a factor in his brain cramp.
"I guess you get a penalty shot for that," he said. "I take full responsibility."
"I've never seen that play by anybody ever," Toronto head coach Mike Babcock said. "We've all got to learn from it."
Danault put Montreal ahead by banging a rebound home with 4:40 left. But Matthews tied it 1:15 to go and Hutchinson on the bench for an extra attacker.
"Things were looking a little bleak going into the second intermission, but we were able to stay positive, rally the troops and have a big third," Price said. "I think that was fun to watch."
The 22-year-old Matthews has 12 goals in 12 career games against Montreal. ...The Leafs came in off Friday's 4-1 road victory in Columbus following their 5-3 victory over Ottawa in the opener. The Canadiens dropped their season opener 4-3 to the Carolina Hurricanes in a shootout Thursday.
Canadiens: At Buffalo on Wednesday night.
Maple Leafs: Host St. Louis on Monday night.
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