The teams traded goals all night but the Lightning ultimately prevailed in overtime when Kevin Shattenkirk scored the game-winner.
Shattenkirk's marker came on the power play after Stars forward Jamie Benn took a tripping penalty. It was the Lightning's third goal of the game with the man-advantage.
The Stars got off to a hot start, scoring the game's first two goals thanks to John Klingberg and Joe Pavelski. The Lightning answered just before the buzzer in the opening frame when Brayden Point burst into the offensive zone and put one past Anton Khudobin.
Point was able to bat the puck out of the air to score again just minutes into the second period to tie it up.
An incredible solo effort by Tyler Seguin led to a goal from Corey Perry, who managed to jam the puck into a wide-open net to put the Stars back on top. But, similarly to the first period, the Lightning netted a power-play marker at the end of the second to tie the game back up.
Alex Killorn got on the board for the Lightning to take the lead minutes into the final frame, but Pavelski scored his second minutes later to tie things up once again.
With the goal, Pavelski became the oldest player in NHL history to score 12 goals in a single postseason and also tied Joe Mullen for the most playoff goals scored by an American skater (60).
Lightning goaltender Andrei Vasilevskiy ended the night with 26 saves on 30 shots, while Khudobin stopped 30 of the 35 he faced.
Game 5 is set for 8 p.m. ET on Saturday.