The Lightning trailed 5-3 in the third period but Steven Stamkos scored his second goal of the game 7 1/2 minutes into the frame, cutting the Bolts' deficit to one.
Then, with only 11 seconds left in the regulation, he fired a pass from behind the net to Teddy Purcell, who converted it for the tying goal.
In overtime, Michel Bergeron's shot went wide but the rebound ricocheted right to Pouliot, who stuffed in the game-winner for his second tally of the contest.
Purcell also added a pair of assists while Anders Lindback turned aside 25-of-30 shots as Tampa Bay snapped a six-game losing streak.
Shawn Matthias and Jack Skille notched a goal and a helper each and Jose Theodore made 30 saves for the Panthers, who dropped a fourth consecutive game.