Fleischmann's power-play tally early in the second period tied it up 1-1 and his empty-netter with 41 seconds remaining gave the Panthers a much-needed insurance goal at 3-1.
In between, Shawn Matthias netted his seventh of the season as first-place Florida maintained a four-point lead over Winnipeg in the Southeast Division.
Jose Theodore started in goal, making 9-of-10 saves, before exiting the game with an injury early in the second period when Panthers defenseman Dmitry Kulikov inadvertently ran into him.
Clemmensen came in and turned aside all but one shot the rest of the way.
Carey Price made 32 saves for the Habs.
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