With the Panthers clinging to a one-goal lead late in the third period, penalties to Mike Weaver and Scottie Upshall gave Montreal a 5-on-3 power play.
But Clemmensen made six straight stops to preserve the lead and secure the Panthers' first three-game winning streak this season.
Jesse Winchester and Nick Bjugstad each registered a second-period goal as Florida improved to 5-1-0 in its last six contests.
Alex Galchenyuk scored Peter Budaj made 23 saves for the Canadiens, who lost for the third time in four games.
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