Michael Eyssimont and Anthony Cirelli each had a goal and an assist, and Andrei Vasilevskiy made 28 saves for the Lightning, the No. 3 seed in the Atlantic Division who lost Games 3 and 4 at home.
"They dug their heels in tonight, and I think who really dug his heels in tonight was Vasilevskiy]," Tampa Bay coach Jon Cooper said. "His name has come up a lot for various reasons over the last couple days, and I think he proved he could handle the high shots."
Morgan Rielly and Auston Matthews scored, John Tavares had two assists and Ilya Samsonov made 34 saves for the Maple Leafs, the No. 2 seed in the Atlantic. They are 0-11 in their past 11 opportunities to eliminate an opponent.
"We're just confident with our group," Rielly said. "We're here for a reason, we play good hockey, we beat them in this series and can do it again. We didn't expect them to roll over, and now it's important for our group to go on the road with a mission."
[RELATED: [Complete Maple Leafs vs Lightning series coverage]
Eyssimont put the Lightning up 2-1 at 4:23 of the second period with his first career Stanley Cup Playoff goal. He skated wide around Justin Holl and shot between Samsonov's pads from the bottom of the left face-off circle. It was Eyssimont's first game since sustaining an upper-body injury in Game 1.
"I had to get around that [defenseman] and he's a great goalie, so sometimes you have to be a little crafty and it's just (a move) I've worked on with (assistant coach Jeff Halpern) at morning skates, so I figured I'd try it," Eyssimont said.
Vasilevskiy preserved the one-goal lead at 5:11 of the third period when he stopped Mitch Marner on a breakaway.
"Ever since I've come to Tampa, each and every night, Vasilevskiy has been our best player," Cirelli said. "He's back there making saves for us, he's a rock for us back there and when he's making those saves, he gives us huge momentum boosts, keeps us in games and gets us going."