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.
Grabovski tied it at 1 on a rebound goal after goalie Anders Lindback got his glove on Nikolai Kulemin's shot from the slot with 2:01 left in the first.
Toronto entered tied for the NHL lead in fighting majors, and right wing Mike Brown fought Tampa Bay forward B.J. Crombeen just 7:56 into the game. Brown also received a 10-minute misconduct.
Both teams received two fighting majors in the third, which included a pair of scuffles.
NOTES: 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.