That was some sigh of relief, which you would expect after 23 years of holding your breath. With one shot in overtime, the Islanders got rid of their playoff hex and took on a whole new identity. The game they won Sunday night didn’t mean the Stanley Cup, but it meant the world to them, their fans and their franchise.

It all happened because of the man to whom it means the world to be an Islander. John Tavares, the captain and player who has done more than anyone in a generation to breathe life into the franchise, tied it in the final minute of regulation and scored at 10:41 of the second overtime to give the Islanders a series-clinching 2-1 win over the Panthers in Game 6 in their most extraordinary night since 1993.

The Islanders all knew that the team had not won a playoff series since that year — representing the longest empty stretch among NHL teams — and they know that playoffs define who you are. Now they can think of themselves differently. But as Tavares pointed out, it wasn’t themselves that they were thinking about after Game 6 of Year 1 at Barclays Center.

“First and foremost, it’s for our fan base. They’ve been dying for this,” Tavares said. “No question, a lot of us haven’t been here that long. But some of us have been here a while. Fransie [Nielsen] I think has been here 10 years, I’ve been here seven, Kyle [Okposo] has been here eight or nine. It’s time. We had to get over this hump.

“Obviously, there’s still a long way to go, but they’ve deserved it,” he said of the people who were ecstatically cheering. “They’ve waited, they’ve been through some tough times. To be part of that is a good feeling right now.”

Yes, the team has done only 1/19th of the amazing run of the early 1980s Islanders, who won 19 consecutive series. There was no champagne. They don’t do that after each round in hockey. It’s just a matter of shaking hands and moving on.

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This time, finally, the Islanders aren’t moving directly into the summer, which meant the world. A major reason why was the cloud had been hovering for so long.

Much has happened to the Islanders since 1993, almost all of it not good. At times, the hockey team seemed only an afterthought or a tool for sealing a real estate deal around Nassau Coliseum. The stretch has been a jumble of jagged edges: the ownership group known as the Gang of Four, John Spano, the Milstein tenure, the notion to hire sumo wrestlers as goalies and other wacky ideas from Charles Wang.

But since Tavares has been here, under general manager Garth Snow and coach Jack Capuano, it has been about the hockey, about building a team. The Toronto-area native would have been a star anywhere, but he immediately and unfailingly embraced Long Island and then Brooklyn. Being an Islander always has meant something to him. And he means everything to his teammates.

“He’s one of the best in the world,” Nielsen said. “He has always been stepping up like that in big moments when we need him.”

Cal Clutterbuck, who played junior hockey with Tavares, said, “Honestly, there are a lot of things that have changed about him over the years, but one thing that has never changed is he wants the puck, he wants to win, he wants to score. Quite honestly, that is something that not everybody has. He has a burning desire to score, to win, to help his team.”

There has been a lot of smoldering desire in the stands for the past 23 years. In an almost mystifying way, the Islanders have grown more popular among young fans as their playoff futility grew year after year.

As a matter of fact, these three home playoff games seemed to change the vibe at Barclays Center. The train rides fostered a new layer of community among them. You could see it in the energy at the Jamaica Long Island Rail Road station yesterday: Dozens of people wearing blue and orange, spontaneously chanting “Let’s Go Islanders,” interspersed with their renditions of the sound effects that are played after Islanders goals.

Yes (or should we say in Islanders lingo, Yes! Yes! Yes!), Sunday night meant plenty to many people. Nielsen said, “It means the world to me.”

He and others had waited a long time for it. Thanks to the captain, their wait is over.