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It was the Lightning who struck first, with Nikita Kucherov taking advantage of an early power play to put the reigning back-to-back Stanley Cup champions ahead 1-0 less than three minutes in.
But the lead would be short-lived, with K'Andre Miller squaring the ledger less than five minutes later, before Kaapo Kakko gave the Rangers a 2-1 lead heading into the second period.
In a clash between arguably the two greatest goaltenders in the game – New York's Igor Shesterkin and Tampa Bay's Andrei Vasilevskiy – the second period was a 20-minute scoreless stalemate, setting up a thrilling third frame.
However, much of the drama was sucked out of the contest when New York's Mika Zibanejad gave the Rangers a 3-1 buffer just two minutes into the period, and Shesterkin was determined to see it out.
A late Nicholas Paul goal set up a frantic final two minutes, but the Rangers' defense held firm, securing the win.
With the win, the Rangers snapped a streak of 17 consecutive Lightning wins following a playoff loss, with their last back-to-back playoff losses coming in April, 2019.
Speaking to the media after the win, Rangers defenseman Adam Fox – who had two assists – said his side are good at playing spoiler.
"We heard all year that [we weren't] really going to have playoff success," he said.
"We’ve said it all year, the belief in the room is high and the outside opinions isn’t really affecting anyone.
"Coming from down 3-1 [against the Pittsburgh Penguins], down 2-0 [against the Carolina Hurricanes] and [the Lightning] obviously had a good playoff streak of not losing back-to-back games, but that’s not really in our minds coming into the games.
"We're not thinking about what streaks teams have or how they've done earlier. It's right now, and we're just trying to bring it day in and day out.
"We did a great job limiting them, especially in the first two periods. We didn't make too many mistakes… [and] when we needed those big saves, we got them at the end, as usual."
Rangers coach Gerard Gallant called it "a huge win".
"It's a huge win for us, but we just get ready for the next one," he said.
"The way we played the last two games, that's the way we're going to have to play to win the series. We want to battle hard, we want to compete hard and we've been a tough out so far.
"We knew they were going to push real hard... [but] we battled, we found a way. We're playing against a real good team over there. They pushed it, we made some key saves at the end."
Lightning captain Steven Stamkos said his side was punished for their mistakes, but he hopes they will carry some momentum into the next fixture back in Tampa Bay after dominating the last 15 minutes.
"We haven’t executed the proper way that got us here," he said.
"They are a skilled team and make you pay. We found some momentum at the end – we have to carry that over."