Tampa Bay also got goals from Steve Downie and Sean Bergenheim. Stamkos has five goals and four assists in five games this season.
Toby Petersen, Adam Burish, Brenden Morrow and James Neal scored for the Stars, who had started the season by winning four in a row. Dallas, which entered with the third-worst penalty-killing unit in the NHL, allowed four power-play goals.
The Lightning were without team captain Vincent Lecavalier, who didn't play because of a right hand injury. He was hit on the first knuckle of the hand by a shot in Saturday's 6-0 loss at Florida.
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.
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.
Notes: Dallas C Brad Richards assisted on Neal's goal. He has 10 points (two goals, eight assists) through five games this season. ... Stamkos, who had just 38 penalty minutes last season, picked up two minor penalties during the opening 8:13 of the first period. ... Stars LW Jamie Benn missed his third consecutive game with concussion-like symptoms. ... Stamkos has three goals and five points in three games against Dallas. ... Tamap Bay's team record for power-play goals in one game is five set at Buffalo on March 19, 1995.