Jose Theodore stopped all three shootout attempts by the Lightning after making 26 saves during regulation and overtime.
The Panthers have won two out of three games this season and they handed Tampa Bay its fourth straight loss.
Florida never trailed, getting a first-period, power-play goal from Ed Jovanovski.
Marc-Andre Bergeron tied it 9:43 into the third, after which Kris Versteeg produced another power play goal for the Panthers with 2:55 to go in regulation. Bergeron scored again 39 seconds later to force overtime.
Mathieu Garon let in the shootout goal to take the loss.