Nylander added an assist for a three-point performance and Auston Matthews, Dryden Hunt and Alexander Kerfoot also scored for Toronto.
''A difference-maker,'' Maple Leafs coach Sheldon Keefe said about Nylander. ''That's more of the type of player he's been for us all season.''
Mitch Marner added two assists to break Darryl Sittler's 1977-78 franchise home points streak record at 19 games.
Matt Murray allowed four goals on eight shots before being replaced by Ilya Samsonov early in the second period. Samsonov stopped 11 shots.
Aleksander Barkov had a goal and an assist and Carter Verhaeghe, Anton Lundell and Josh Mahura also scored for Florida. Sergei Bobrovsky made 32 saves.
''There will always be penalties you take that you earn . and there's a whole bunch you're not going to like,'' Panthers coach Paul Maurice said. ''I thought we had the inordinate share of those.''
Nylander missed the net on a penalty shot with Toronto trailing 4-3 early in the third period, then scored at 5:59 after Marner's initial shot hit Bobrovsky's pad and caromed in off the Toronto forward.
''He's got the reserve tank like I've never seen,'' Marner said about Nylander. ''Finds ways out there, especially in overtime, to give himself space. Turns on the burners.''
Nylander then buried the winner in the extra period, showing great patience on a terrific individual effort before sliding the puck past Bobrovsky.
Panthers: At Montreal on Thursday night.
Maple Leafs: Host Winnipeg on Thursday night.
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