"For me, the most important thing was getting the two points," said Nashville defenseman P.K. Subban, who played his second game in Montreal since he was traded to the Predators on June 29, 2016.
Brendan Gallagher and Jonathan Drouin scored, and Carey Price made 30 saves for Montreal (22-26-7), which has lost two straight and is 4-6-3 in its past 13 games.
With Price pulled for an extra skater, Drouin tied the game 2-2 with 1:14 left in the third period on a slap shot from the left point that went over Rinne's right shoulder on the short side.
"All I can say is we deserved a better result," Canadiens coach Claude Julien said. "I think we deserved to win that game. We did everything, and tonight there were two good goalies, and that kept the game tight. But I have nothing negative to say about my team, because from start to finish, we played a solid game. We did everything we could to win, and we should have won."