The two teams went to a shootout after battling to a 3-3 tie and going scoreless in a 5-minute overtime.
Valtteri Filppula scored for the Lightning and Brad Boyes answered for Florida before Stamkos beat Jacob Markstrom with a forehand shot in the fourth round for the game-winner.
Radko Gudas and Martin St. Louis scored in regulation and Anders Lindback made 29 saves for the Bolts, who won a third straight game.
Boyes and Nick Bjugstad each scored in the third period for the Panthers, who erased a late 3-1 deficit to force overtime.
Shawn Matthias scored and Markstrom finished with 25 stops.