The captain carried his team to the second round.

John Tavares’ determined solo effort to grab his own rebound, circle the net and sweep it home with 9:19 left in the second overtime secured a 2-1 Islanders win and a 4-2 first-round series victory over the Panthers Sunday night, the first playoff series win for the franchise since 1993.

The Islanders will face the Tampa Bay Lightning in the second round, with Game 1 in Tampa later this week.

Tavares got the game to overtime with a rebound sweep past Roberto Luongo with 53.2 seconds left in regulation to bring a restless, sellout Barclays Center crowd alive after 59 frustrating minutes. Just over half an hour of extra game time later, they were delirious while celebrating their franchise player’s winning goal.

“It was fitting,” Jack Capuano said of Tavares. “He gets the tying goal, then the winner on a great individual effort . . . This is a resilient group.”

It’s a core that has been together for many years now that has experienced several seasons when the playoffs were a pipe dream and two first-round exits that were bravely fought but fell short.

This spring, overtime was the right time. Thomas Hickey ended Game 3 in the first OT in the first-ever Barclays Center playoff game. Alan Quine, the little-known rookie, ended Game 5 in Sunrise after 96 minutes to give the Islanders a 3-2 series lead.

Game 6 was prime time for Tavares, who ended the series with five goals and nine points, most on either team. He took a feed from Kyle Okposo and snapped a wrist shot that Luongo stopped, the 49th save of the game for the Panthers veteran goaltender.

But Tavares drove the net for the rebound, got between Florida defensemen Brian Campbell and Aaron Ekblad, circled the Panthers’ net without Luongo tracking him and deposited a backhand into the yawning far side to kick off a jubilant Isles celebration.

“Just with the way things have gone for the franchise, a lot of frustrating seasons, you understand that passionate fans get restless,” Cal Clutterbuck said. “The bar was set pretty high for this organization a lot of years ago. This feels pretty good for everyone.”

It was nearly a massive letdown. The Islanders played their best opening period of the series, but the Panthers were clogging shooting lanes and managed to grab the lead for the fifth time in the six games when Jonathan Huberdeau beat Thomas Greiss on a deflected shot at 18:58 of the first.

Greiss, who finished with 41 saves, held the deficit to a goal with a solid second period. The Islanders made some minor forays through the third but weren’t getting through.

With Greiss off for an extra skater, the Panthers’ Vincent Trocheck missed the empty net with just over a minute to go. Nick Leddy retrieved the puck and went end to end, with his sharp-angle backhand deflecting off Luongo into the slot.

Nikolay Kulemin slid a shot that Luongo stopped but he didn’t know that the puck had squirted free. It sat in the crease. Tavares saw it and scored to tie it and get the game to yet another overtime.

Once there, the Islanders pressed in the first 20 minutes, with Quine denied on a semi-breakaway and Luongo smothering Brock Nelson’s two-on-one try.

“I thought that [first OT] was the best period we played all series,” Okposo said.

Reilly Smith lifted a shot off the crossbar early in the second OT but that was the most threatening chance on either side as the minutes wore on, until Tavares ended it.

“Once he went around the net I knew it was over,” Okposo said. “It was just . . . elation. We made three huge plays in the series in OT and the biggest was tonight. To do it in front of our fans, play the way we did, I’m extremely proud of our group.”

It’s a group that gets to go one more step than it’s been since before a few of the current Islanders were born. The 23 years doesn’t weigh as heavily on their heads as does the seven years of Okposo, Tavares, Frans Nielsen, Josh Bailey and Travis Hamonic being together through the few ups and many downs.

“It feels great to do this with those guys,” Okposo said. “We grew up here together. To break through that invisible barrier that’s been hanging over this franchise for the last 23 years, it’s special. But it’s also just one chapter in a book we haven’t finished yet.”