With the game even at 1, Purcell carried the puck from the right circle into the slot and backhanded a shot past goalie Kari Lehtonen at 3:36 of the final period for Purcell's 11th of the season.
Stamkos had 11 goals in his previous 11 games and leads the NHL with 31. All five of his shots were turned aside by Lehtonen, but Purcell gave his team a lift.
Martin St. Louis had the primary assist on each of Purcell's goals as the Lightning ended a six-game winless streak on the road (0-5-1).
"[Stamkos and St. Louis] carry the load night in and night out so it was nice to provide some secondary scoring and chip in where I can," Purcell said.
The Lightning have allowed the most goals in the NHL this season (160) and had given up three or more goals in each of their last seven games.
But Garon and the Tampa Bay defense shut down a Dallas offense that has managed four goals in the last four games.
"We've played a lot better defensively," Garon said. "That makes it easier on everybody. When you give up four, five, it's impossible to win. We've sorted out a couple of things. When there's a little mistake, we're standing up for each other. We've had more confidence the last two games."
The Lightning have won two straight following a seven-game winless streak (0-6-1), having beaten Boston 5-3 on Tuesday night.
"It's two good wins," Purcell said. "It was a big emotional win to beat the Bruins. We wanted to put a streak together and came out strong."
Tom Wandell scored and Lehtonen stopped 20 shots for the Stars, 1-4-1 in their last six.
Purcell was alone in the slot after a feed from St. Louis and flipped a wrist shot past Lehtonen with 1:12 left in the first period.
"Marty made a joke that I had a 3-seconds-in-the-paint violation," Purcell said. "He made a world-class pass and I kind of waited [Lehtonen] out."
The Lightning dominated the first period with a 13-5 shots advantage.
"We've been focusing on our starts, specifically on the road," Tampa Bay coach Guy Boucher said. "We had a lot of shots in the first period and didn't give up a lot."
Garon was down for several minutes in the crease at 4:33 of the second period with an apparent right shoulder injury after he collided with teammate Steven Stamkos on a scramble in front of the net, but Garon remained in the game.
Garon said he had a stinger, but it didn't affect him the rest of the way.
Dallas drew even at 1 at 1:22 of the third period when Loui Eriksson's centering pass glanced off Wandell's skate and past Garon. The goal was upheld when a video review showed that Wandell didn't kick the puck into the net.
But the Stars were unable to get much going on offense all night despite a 28-22 edge in shots, paying for a lethargic start.
"The frustrating part was the first 30 minutes of getting outworked," Stars coach Glen Gulutzan said. "....Some team comes into your building on a Friday night and outplays you for 30 minutes, I got a real problem with it. We have to work 60 minutes every night with everyone in our lineup if we want to make the playoffs. If we don't, we won't play. That needs to be learned."
- Tampa Bay is without D Victor Hedman (concussion), D Marc-Andre Bergeron (back), RW Adam Hall (arm), D Mattias Ohlund (knee), RW Ryan Shannon (lower body) and RW R.T. Wyman (finger).
- The teams were meeting for the only time this season.
- Dallas is 4 for 45 on the power play in the last 14 games. Tampa Bay is 2 for 27 with the man advantage in its last nine games.