Mathew Barzal and Ryan Pulock scored for the Islanders, who lost in six games to the Lightning in the Eastern Conference Final last season, including 8-2 in Game 1.
"I thought today was more reminiscent of Game 2 and 3 and 4 and 5 and 6 from last season," New York coach Barry Trotz said. "I think teams are pretty well the same as they were last year. Systematically, we're pretty similar. There hasn't been any overhaul, I don't think, on both sides.
"Pretty well what we expected, and in some ways that was good. Being familiar with a team that you haven't played for [nine] months, it was good. It was a good start for us."
Brayden Point scored, and Andrei Vasilevskiy made 29 saves for the Lightning, who are trailing in a series for the first time this postseason.
Teams that take a 1-0 lead are 497-226 (.687) winning a best-of-7 Stanley Cup Playoff series, including 8-4 this season. Game 2 is at Tampa Bay on Tuesday (8 p.m. ET; NBCSN, CBC, SN, TVAS).
"Our work ethic was there, our compete was there, our minds weren't there," Lightning coach Jon Cooper said. "Some of our decisions were poor. That's what happens. You get this deep into the playoffs, there are only four teams left, you've got to have everything working in unison, and we just weren't quite there tonight."
Barzal gave New York a 1-0 lead at 12:32 of the second period. Josh Bailey intercepted a pass from Steven Stamkos at the Lightning blue line and sent a stretch pass to Barzal, who shot between Vasilevskiy's pads.
Barzal has four goals in five games after not scoring in the first eight of the playoffs.
"In some games we've played, I don't need to get the fifth goal when we're up 4-1," Barzal said. "It's just a matter of trying to be there for the boys when they need me and just doing the right things."
Varlamov kept New York in front with a save on Alex Killorn's backhand during an Islanders power play at 18:27 of the second period. He also made three saves during a New York man-advantage in the first.
Killorn was denied again when Varlamov made a right-pad save on his shot from the slot at 3:47 of the third period.
Pulock, who scored two goals in 56 regular-season games, then scored his fourth of the playoffs with a slap shot from the point at 5:36 to make it 2-0.
Three of Pulock's four goals have been game-winners.
"I tried not to let that bother me in the regular season and just keep playing my game, play the right way, play good defensively," Pulock said. "I knew it would come, but obviously come playoff time, it's nice to contribute in that way and kind of help out offensively. Every goal is huge in the playoffs. It's nice to get one or two, just want to keep that going."
Point scored to make it 2-1 and end Varlamov's shutout bid with 54 seconds remaining with a 6-on-4 goal after Islanders forward Brock Nelson was assessed a high-sticking penalty against Nikita Kucherov and Vasilevskiy was pulled for the extra skater.
Tampa Bay's top line of Ondrej Palat, Point and Kucherov combined for three shots on goal. Point had a chance to give the Lightning the lead but shot over the net on a breakaway at 12:26 of the first period.
"It's weird because we're playing some really contrasting style teams," Cooper said of New York and the Carolina Hurricanes, who Tampa Bay defeated in five games in the second round. "The Islanders do what they do, and they do it extremely well. We saw it, so now it's on us to pick it apart."
Vasilevskiy made back-to-back pad saves against Jean-Gabriel Pageau and Kyle Palmieri at 9:14 and 9:21 among his 14 in the first period.
"We knew it was going to be extremely tough," Stamkos said. "We've been in this position before, so there's definitely no panic in this room. But we do realize that it's going to be an extremely tight series and we just have to expect that going forward."
NOTES:Varlamov and Ilya Sorokin each has won four straight starts for the Islanders this postseason. It's the third time in NHL history that a team has had two goalies win at least four starts in a row in the playoffs (Eddie Johnston and Gerry Cheevers with the Bruins in 1972; Phil Myre and Pete Peeters with the Philadelphia Flyers in 1980). … Tampa Bay is 3-3 at home this postseason. … Islanders forward Oliver Wahlstrom participated in warmups for the first time since sustaining a lower-body injury in Game 5 of the first round against the Pittsburgh Penguins.
