The win came just two days after the Lightning were blown out in a 6-0 loss at Florida. Tampa Bay outshot Dallas 44-19, including 35-10 through two periods.
"I think everybody rebounded," Tampa Bay coach Guy Boucher said. "We came out real strong."
Tampa Bay, which went 4-for-5 on the power play, also got goals from Steve Downie and Sean Bergenheim. Stamkos, who led the League with 51 goals in 2009-10, has five goals and four assists in five games this season.
"The focus every game right now is on improvement, whether we win or lose," Moore said. "I thought tonight was one of our most complete efforts of the season."
Toby Petersen, Adam Burish, Brenden Morrow and James Neal scored for the Stars, who had started the season by winning four straight. Dallas, which entered with the third-worst penalty-killing unit in the NHL, has given up 10 power-play goals in 25 shorthanded situations.
"We couldn't kill a penalty to save our life," Stars coach Marc Crawford said.
Tampa Bay took a 3-2 lead when Bergenheim scored from the top of the crease off a pass from Stamkos with 10:34 remaining in the second. Moore extended the advantage to 4-2 on a power-play goal at 18:06 of the second.
Petersen put the Stars up 1-0 with a short-handed breakaway goal at 15:04 of the first. It stopped a personal 17-game goal drought, dating back to March 16th against San Jose.
After Downie tied it at 1 during a power play at 16:21 of the first, Stamkos’ man-advantage goal gave Tampa Bay a 2-1 lead with 13.2 seconds left in the period.
"You've got to get shots on goal," Dallas center Brad Richards said. "We didn't test their goalie. If we get 30 shots, maybe we win."
Burish pulled the Stars even at 2 from the slot 6:04 into the second.
Moore, on the power play, and Morrow traded goals 13 seconds apart early in the third. Dallas made it a one-goal game when Neal scored with the Stars’ net empty and 59.5 seconds to play.
"We were going in the right direction from the start," Tampa Bay defenseman Victor Hedman said. "It got a little close at the end there, but overall I think we dominated the game."
The Lightning were without captain Vincent Lecavalier, who didn't play because of a hand injury. He was hit on the first knuckle of the right hand by a shot on Saturday.
Material from wire services and team media was used in this report.