Ryan Lindgren scored, and Igor Shesterkin made 24 saves for the Rangers, the No. 2 seed in the Metropolitan Division, who had won a team-record eight straight Stanley Cup Playoff games at home.
"It's tough tonight," New York coach Gerard Gallant said. "It's a tough night, there's no doubt. But tomorrow we'll get on the airplane and we'll be ready to go play a game in Tampa Bay. It'll be a battle. They're a good hockey team. That's why they're the Stanley Cup champs. We're going to have to go in there and play our best game and win a game to give ourselves a chance for Game 7 back here."
Lindgren put the Rangers up 1-0 at 10:29 of the second period when he flung the puck on goal from the left boards near the face-off circle and beat Vasilevskiy to the short side.
"We just couldn't seem to get that next goal," Lindgren said. "We've been in this situation before and it seems like we play our best hockey when our backs are against the wall. We've just got to go to Tampa and win a game. Take it one game at a time. Draw from those past experiences and go play our best hockey."
Sergachev tied it 1-1 at 17:34 with a shot from the point through traffic. He hadn't scored in 39 playoff games dating to Game 5 of the 2020 Cup Final against the Dallas Stars.
"That one went in and it's a good feeling," Sergachev said. "I haven't scored a lot of goals, but it definitely felt great."
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Lightning defenseman Victor Hedman said, "It was huge for us and you need that in the playoffs, guys to come through, and Sergie has been playing great hockey and I'm glad he got on the score sheet. Two great shots like 'Pally' said. Perfect wristers in the perfect spot. … Two great shots and overall a great game."
Brandon Hagel scored an empty-net goal with 59 seconds remaining for the 3-1 final.
It was the third straight game Palat scored, and second time in three games he had the game-winner in the final minutes. He scored with 42 seconds remaining in a 3-2 win in Game 3 and has four goals in the series.
"'Pally' has been a guy that, like I've talked about and we've all talked about, is just the warrior mentality out there," Stamkos said. "He's willing to do anything and he gets rewarded because he plays the game the right way. Whether it's hockey gods or karma or whatever it is, you do the right things, things are bound to happen."
Tampa Bay has allowed four goals in the past three games after allowing nine in the first two.
"We didn't like our first two games for sure," Stamkos said. "Very uncharacteristic plays and just sloppy hockey. … We always talked about you give up one or two, you should be able to win those games. We found a recipe here."