Nielsen zipped a low forehand shot past Ben Bishop on the Islanders' first attempt in the tiebreaker. Poulin denied Nikita Kucherov and Victor Hedman in the second and third rounds to give the Islanders their fourth win in five games on a six-game trip that ends Saturday night against the Philadelphia Flyers.
Poulin, 23, has played all five games on the trip and is 5-1-0 since 38-year-old starter Evgeni Nabokov went down with a lower-body injury early in a 7-3 win against the Dallas Stars on Jan. 6.
"Kevin was solid," coach Jack Capuano said. "He really controlled his rebounds. He was really poised back there and he made some timely saves. With Nabby hurt, this is an opportunity for Kevin and hopefully he'll keep playing the way he's been playing for us."
Poulin improved to 3-0 in shootouts this season and has stopped nine of 10 attempts.
"It's nice to see the way he has stepped up," Nielsen said. "He has had some solid saves on this road trip."
It was a measure of revenge for the Islanders (19-23-7), who lost 3-2 to the Lightning in a shootout at Nassau Coliseum on Dec. 17.
Martin St. Louis scored a first-period power-play goal for the Lightning (28-15-5). Thomas Vanek tied the game with 1:38 left in the second by scoring with Hedman in the box for high-sticking Brian Strait.
The Lightning carried the play for most of the third period, at one point outshooting the Islanders 7-1, but were unable to beat Poulin. New York's best chance in the final period came when John Tavares andKyle Okposo had a 2-on-1 break with five minutes remaining, but Richard Panik disrupted Tavares' pass with a diving stop.
St. Louis extended his scoring streak to six games when he opened the scoring at 5:45 of the first period with Strait in the penalty box for delay of game after he sent the puck into the stands. Valtteri Filppula, standing outside the crease to Poulin's right, sent a pass across the crease to St. Louis, who flipped the puck into an open net for his 21st goal of the season. Tyler Johnson also was credited with an assist.
"They had their chances; we had chances," St. Louis said. "Both goalies played well. It could have gone either way."
Thanks to Poulin and Nielsen, the Islanders left with the extra point.
"In all seriousness, I thought we played pretty well tonight," Lightning coach Jon Cooper said. "We gave up one goal and we should get a point out of that, if not win the game. We got one point, we didn't get two. That's the tough part. But we battled. We played hard; we had every chance to win the game, we just didn't. We played a solid defensive game, the puck just didn't go in for us tonight. It was an even game."
The Islanders had the better of play in the second period and finally tied it when Vanek took a pass from Nielsen in the low slot and fired the puck past Bishop for his 17th goal of the season. Vanek's goal came five seconds into Hedman's penalty and was the Islanders' first power-play goal in four games.
"I knew Vanek was going to go to the slot," Nielsen said. "We knew they dropped their defense behind the center so they would only have one guy in the front of the net."
The Lightning has at least one point in each of Bishop's last 12 games. He matched Poulin with 29 saves through 65 minutes.
"We were battling since they scored the go-ahead goal," Poulin said. "But we were able to come back and play hard. We did the little things that made us successful."
They made an impression doing as well.
"They are a tough team to beat," Hedman, playing in his 300th career NHL game, said. "It was a tough game."