Tampa Bay was unable to generate much offense in the first half of the game, and again had trouble limiting possession and chances by New York as it did in Game 1, a 6-2 loss.
"I just think we haven't executed the proper way that has got us here, especially the last couple of years in terms of puck management and execution, and stuff like that," Lightning captain Steven Stamkos said. "We talked about the amount of skill they have on their team. If you give them chances, they are a very skilled team and they can make you pay."
Paul brought Tampa Bay within 3-2 with 2:02 remaining and Vasilevskiy pulled for an extra skater.
The Lightning outshot the Rangers 14-6 in the third period, including 3-0 in the final 59 seconds.
"We got better as the game went on," defenseman Victor Hedman said. "We had a bunch of opportunities to tie the game up there at the end, so we'll take that with us going into next game. We can't hang our hats just because we are going to home ice, but it's going to be great to be in front of our fans and try and get the next game."
Shesterkin made several big saves in the final minute, including one on Hedman with 22 seconds remaining.
"I can't imagine [how many] times people have said nice things about Igor bailing us out at this point in the year," Miller said. "[Shesterkin is] unreal. You see it every game, the number of highlight-reel saves. Obviously we have a lot of trust in him and try to let him do his thing. We try to limit the chances that he sees, block shots and doing those little things, but 'Shesty's' Shesty, he's going to stop the puck."
Vasilevskiy has allowed nine goals on 62 shots in the two games in the series; he allowed three goals on 154 shots in a four-game sweep of the Florida Panthers in the second round.
"I still stand by my opinion," Shesterkin said. "Everything is as I said before: Andrei is still the best goalie in the world. You can't forget that the series is four wins, so you just have to keep playing"
NOTES: Zibanejad extended his home goal streak to six games. Only three players have a longer streak in a playoff year (Guy Lafleur, eight games in 1979; Paul Coffey, seven games in 1985; Reggie Leach, seven games in 1976). … Zibanejad has 11 points (six goals, five assists) in a seven-game point streak. … Fox has 13 points (three goals, 10 assists) in nine home playoff games this season. The only Rangers players with more in a single postseason are Mark Messier and Brian Leetch, who each had 16 points in 13 home games in 1994, and Sergei Zubov, who had 13 points in 13 games in 1994. … New York won a game for the first time in these playoffs without scoring a power-play goal; they had scored with the man advantage in each of their previous nine wins (12-for-30, 40.0 percent). … Lightning forward Corey Perry played his 180th playoff game to tie Sidney Crosby for third among active players behind Zdeno Chara (200) and Joe Thornton (187).