The win, New York's sixth in seven games, avenged its only loss in that span; Tampa Bay defeated the Islanders 5-2 on Saturday.
"We did the smart things," said Islanders center Casey Cizikas, whose breakaway goal 8:55 into the third period completed the scoring. "We got pucks behind their 'D', made them turn and skate every time. That was the biggest difference."
This time, the Lightning were the team playing on consecutive nights, just as the Islanders were three days ago.
"It's a tough spot, coming off a back-to-back," Tampa Bay coach Jon Cooper said. "They were a hungry team; hadn't played in a few days."
Tampa Bay led 1-0 after one period, but the Islanders (12-6-0) stepped up their play in the final 40 minutes.
"It seemed like they were maybe a little bit faster, a little quicker," said Nabokov, who spent the past three seasons with the Islanders before signing with the Lightning during the summer. "They moved the puck very well on the power play and played well around the net."
The 39-year-old stopped all 12 shots he faced in the first period, but allowed three goals on 12 shots in the second, as well as third-period goals by Brock Nelson and Cizikas. He finished with 31 saves.
Both teams felt the turning point came early in the second period.
The Lightning led 1-0 and began the period on a power play. But Clutterbuck picked off a pass by Anton Stralman outside the Lightning blue line, raced into the right circle and fired a high shot from the faceoff dot that beat Nabokov to get the Islanders even.
"Maybe that first goal gave them a boost," Nabokov said. "We killed a 5-on-3, a 5-on-4, then we got a power play and gave up a goal. After that, the game changed and they got momentum and kept coming. We gave up the second goal and then we were chasing a little bit."
Islanders coach Jack Capuano agreed that Clutterbuck's goal got New York headed in the right direction.
"The Clutterbuck goal was huge," he said. "We were a little bit flat and the penalty kill came up big for us. He brings energy, he's a smart hockey player, he's got some experience and he plays the game the right way."
Jaroslav Halak, who replaced Nabokov as the Islanders' No. 1 goaltender, made 26 saves. He allowed goals by Alex Killorn in the first period and Ondrej Palat in the third but earned his fifth consecutive victory.
Tampa Bay (13-5-2) had won its previous two games, including a 5-1 victory against the New York Rangers at Madison Square Garden on Monday. The Lightning were also reduced to 11 forwards when Cedric Paquette left in the first period because of a lower-body injury and didn't return.
"Everybody knew they were going to be a little tired after playing [Monday] night," Halak said. "In the first period, even though we were down, I think we played pretty well. We came out hard in the second period and got some goals."
The Lightning killed off 67 seconds of a 5-on-3 Islanders power play before taking the lead at 11:53 of the first period. Halak stopped Jason Garrison's shot from the left point and denied Killorn on the rebound. But Killorn was able to get another whack at the puck and slipped it through Halak's legs and into the net for his fourth goal of the season and second in as many nights.
The Lightning started the second period with 1:53 of power-play time, but the Islanders tied the game on Clutterbuck's goal, his second of the season and New York's third while playing shorthanded.
Grabovski put the Islanders ahead for the first time at 12:23 with one of the easiest goals he'll ever score. Kyle Okposo got the puck to Frans Nielsen, who faked a shot, drew the defense and Nabokov toward him and fed Grabovski in the lower right circle. Grabovski put the puck into a vacated net for his fourth of the season.
It became 3-1 at 13:26 when Strome took a pass from Calvin de Haan in the high slot; with Josh Bailey setting a screen, Strome's wrist shot hit the stick of a Lightning defenseman and sailed past Nabokov for his third goal of the season.
"We knew we had to be a little better," Halak said of the Islanders' play in the second period. "I think everybody stepped up and we played a really good second period."
Nelson deflected Lubomir Visnovsky's straightaway 55-foot slap shot past Nabokov 36 seconds into the third period for a power-play goal and a 4-1 lead. Palat scored at 2:14, but Cizikas roofed a breakaway at 8:55 to restore New York's three-goal margin.
"As a group, we stayed calm and poised and persistent," Islanders captain John Tavares said. "We have a good group that feeds off one another. Everyone stays with it, and you can see the depth that we have; guys stepping up all the time. That gives a big lift to everybody."
Nabokov got some applause when his name was announced as the Lightning's starter, and fans broke into a chant of "Nabby" after play resumed following Cizikas' goal.
"I think they were cheering for the wrong reason," he said with a smile. "I had a good time here. The fans were good. Last season we obviously didn't make the playoffs, but the season when we did (2012-13), they were great. When you win, everything goes well. When you lose, everything goes the other way."