Vincent Lecavalier, Alexander Killorn and Matthew Carle also scored for the Lightning, who have won two in a row after an 0-5-1 skid.
Toronto goalie Ben Scrivens had his bid for a third consecutive shutout thwarted midway through the first period on Lecavalier's goal.
"Hot goalie or not, every night is a different night," Lecavalier said. "We played patient and it paid off. We took advantage of they played last night."
Scrivens keyed the Maple Leafs' 3-0 win at Florida on Monday.
"It seems like I was getting some fortunate bounces the last couple of games, and today they bounced the other way," Scrivens said. "That's the nature of the game. Those are the breaks."
"I don't know if we demonstrated enough determination or desperation until the last 7 minutes of the hockey game," Toronto coach Randy Carlyle said. "It seemed like we were a flat group for a good period of the game, but at one point I looked up, it was 2-1 after two periods, and we'd only given up 10 shots. If it wasn't for the lopsided number of power plays, I don't know if we would have given up 15 shots in the game."
The Maple Leafs, 0 for 3 on the power play, outshot Tampa Bay 25-19. The Lightning converted one of six man-advantage opportunities.
"They've got a lot of skill, so they can make a ton of plays happen," van Riemsdyk said. "They're a good team, and they were the better team today."
Stamkos put the Lightning up 2-1 from just outside the crease off a pass from Cory Conacher at 4:56 of the second.
Killorn made it 3-1 when his bouncing shot trickled past Scrivens 2:49 into the third. Carle ended Scrivens' night on a power-play goal with 13:02 to play.
"We're battling. That's the thing that got away from us the six games we lost in a row," Pouliot said. "Everyone played hard, and we come out with wins if we play that way."
Jussi Rynnas replaced Scrivens, and van Riemsdyk cut the Toronto deficit to 4-2 late in the third.
"We played 55 solid, solid minutes," Tampa Bay coach Guy Boucher said. "I didn't like the last 5 minutes."
Lecavalier opened the scoring at 9:25 of the first on a wrist shot from the low left circle that stopped Scrivens' shutout streak at 154 minutes, 16 seconds.
Both teams received two fighting majors in the third, which included a pair of scuffles.
Stamkos has nine goals and 20 points in 17 games against the Maple Leafs. ... Toronto G James Reimer missed his fourth straight game because of a knee injury. ... Lecavalier has two goals and five points during a five-game point streak. ... Grabovski stopped an 11-game goal drought. ... Carle's goal was his first against Toronto, in his 19th game.