Anthony Cirelli also had a goal and an assist, Nicholas Paul scored and Alex Killorn got an empty net goal for Tampa. Andrei Vasilevskiy made 28 saves.
Morgan Rielly and Auston Matthews scored for Toronto, which got 34 stops from Ilya Samsonov. John Tavares had two assists.
The Leafs still lead the best-of-seven series 3-2.
Game 6 will be Saturday in Tampa. Game 7, if necessary, would be Monday back at Scotiabank Arena.
“It’s a game of inches this time of year,” said Rielly, who scored five minutes into the first period to give the Leafs an early lead.
Toronto, which has failed to advance in the post-season since 2004 and was eliminated by the Lightning in seven games in the first round last year, is now 0-10 since 2018 with a chance to eliminate a post-season opponent.
“Didn’t get the result,” Toronto coach Sheldon Keefe said. “But I don’t think it has anything to do with the moment or anything like that. It’s a good hockey team we’re playing.
“They played as perfect a road game as you could probably ask.”
Tied at 1 after a spirited first period, the Lightning pushed ahead at 4:23 of the second when Eyssimont beat Samsonov through the pads from a tough angle for his first goal of the series.
“He’s almost one-on-one with me,” Samsonov said. “Good shot … good goal.”
The period ended in controversy when Pat Maroon crushed Toronto defenseman Mark Giordano from behind into the boards. The initial play went uncalled, but the bruising Tampa winger was assessed a roughing penalty for the ensuing scrum.
Giordano rejoined his teammates on the bench shortly after the restart after getting looked at by the trainer.
Paul made it 3-1 at 11:53 of the third with his first after the Leafs goaltender got bumped by teammate David Kampf.
The home side pulled within one with 3:34 left in regulation when Matthews banged home a loose puck for his fourth, and third in the last two games, with Samsonov on the bench for an extra attacker.
The Leafs continued to press desperately looking for the tie and send the teams to a third straight overtime before Killorn sealed it with five seconds left.
“Don’t bet against the guys - not that group,” Lightning coach Jon Cooper said. “I’m not saying we’ve won the series. We’ve won one game and still have so far to go.
“They dug their heels in tonight. And who really dug his heels in tonight was the goalie.”
Toronto, which has lost six straight series since 2017, opened the scoring at 4:46 of the first when Rielly - the longest-serving Leaf - scored his third of the series from the slot after Tavares beat three Lightning players to the puck to ignite the towel-waving crowd inside a frothing Scotiabank Arena.
Coming off Stanley Cup triumphs in 2020 and 2021 before falling in last year’s final to the Colorado Avalanche, and looking to keep their season alive, the Lightning responded just 26 seconds later when Cirelli scored his third off a scramble.
The Lightning forward suited up for the first time since getting knocked out of Game 1 on a hard hit by Leafs defenseman Jake McCabe. Eyssimont’s inclusion meant Tanner Jeannot watched from the press box.
The gritty Toronto native was eligible to return to the lineup after serving his three-game ban, but Keefe elected to go with the same lineup for the fourth straight game. Michael Bunting registered 23 goals and 49 points in the regular season.
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