Jeff Petry had four assists for Montreal (24-22-7), which has won two straight games and six of eight.
It was Price's 47th NHL shutout, moving him past Ken Dryden for third in Canadiens history.
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"It's pretty cool," Price said. "It's special for me because I had a conversation with Ken, I think it was my third season in the League when things weren't going that well, and he helped me through it. So I'm very grateful for that and I'm looking forward to keep on going."
Sergei Bobrovsky made 31 saves for Florida (28-17-5), which played its first game since Jan. 21. The Panthers are 13-5-0 in their past 18 games.
"They came ready to play," Florida coach Joel Quenneville said. "Right off the bat, they were engaged physically, they won all the loose-puck battles, were first to pucks, and were much better in front of our net, and we were not good in front of their net."
Nick Suzuki gave Montreal a 1-0 lead with his 10th goal at 5:25 of the first period. He scored on a wrist shot into the top right corner after Joel Armia relayed Petry's breakout pass from the Canadiens zone.