Jeff Halpern's fifth goal this season created a 3-3 tie at 16:30 of the third period.
After a spirited but scoreless overtime, Stamkos scored the game-winner in the shootout's first round.
Martin St. Louis had a goal and an assist in regulation. Cory Murphy (power play) also scored, while Ramo was credited with 34 saves on 37 shots for the Lightning, who have split their last four outings.
Florida's David Booth scored two goals, one on the power play, and Michael Frolik scored once. Tomas Vokoun yielded allowed three goals on 27 shots in the Panthers' third straight setback.