Mathew Barzal scored his sixth goal in eight games, and Cal Clutterbuck and Adam Pelech each had two assists for the Islanders, who have won Game 4 to tie each of their three best-of-7 series in the Stanley Cup Playoffs. Varlamov made 28 saves.
"Our game was good," New York coach Barry Trotz said. "I know that we probably had more chances when I look at the game. … Our 60-minute game was pretty decent other than the seven or eight minutes where they surged on us, and we had a surge earlier.
"This is what we expected, 2-2 going back (to Tampa). It's a best-of-3 now."
Brayden Point extended his goal streak to seven games, and Andrei Vasilevskiy made 27 saves for the Lightning, who lost for the second time in eight road games.
Game 5 is at Tampa Bay on Monday (8 p.m. ET; NBCSN, CBC, SN, TVAS).
"It was red rotten, that second period," Lightning coach Jon Cooper said. "These guys have given it their all for whatever it's been now, a month and change, and sometimes you lay an egg, and we laid one in the second period. It was unfortunate because you don't play well sometimes, the breaks don't go your way.
"… You realize how precious every period is and every shift is and it's one of those, if you do bend, you can't break. And that second period, we broke. It was a little self-inflicted. I'm proud of the guys for the push in the third, and you know what? We're one save away from probably still playing right now. But you can't play 40 (minutes), you've got to play 60."
Varlamov made a save on Nikita Kucherov's shot from the slot with 5:36 remaining to preserve the one-goal lead, before Pulock prevented the tying goal.
"The situation that we're in and how deep we are in the playoffs and how important these games are, obviously there's probably no situation like this that I've had before," Pulock said. "Obviously it feels good to score goals, but when you can save a game like that and get the win like that, it's a good feeling."