Stars forward Corey Perry's second score was the game-winning goal.
Perry opened scoring with minutes remaining in the first period, giving the Stars the lead. Lightning forward Ondrej Palat got Tampa on the board in the second, tying the game up heading into the third.
Just over three minutes into the final frame, Lightning defenseman Mikhail Sergachev beat Stars goaltender Anton Khudobin to score the go-ahead goal. Stars veteran Joe Pavelski tied things up with 6:45 remaining in the third period and forced overtime for the second straight game.
Pavelski's 61st career goal sets the record for most playoff goals scored by an American-born player, passing Joe Mullen.
Khudobin was superb in the game, stopping 39 of the 41 shots he faced. Andrei Vasilevskiy let in three goals on 33 shots.
Dallas now trails the series 3-2. The teams will play Game 6 on Monday at 8 p.m. ET.