Evgeny Kuznetsov scored in regulation and the only goal in the shootout, and goaltender Braden Holtby made 35 saves plus three stops in the tiebreaker for Washington, which remains in contention for an Eastern Conference wild card into the Stanley Cup Playoffs.
"I don't think there's any mental relief, we have a hefty job to do still," Holtby said. "So I think that's one step that we needed to do, and we have to keep building, have to keep finding a way to win games like we did tonight."
It was the Capitals' fifth shootout in their past eight games; they have won twice. Washington (35-30-13) is two points behind the Columbus Blue Jackets in a five-team wild card chase with four games to play.
"I think the only thing we can worry about is ourselves," Capitals center Nicklas Backstrom said. "We have to try to win all the games that we have left. We can't control anything else. Whatever happens with the other teams, if they win or lose, we can't control it. Just focus on us."
The Islanders have not lost in regulation in their past seven games (5-0-2).
Some fortunate bounces allowed the Capitals to tie the game 3-3 in the second period.
Alex Ovechkin misfired at an open net with goalie Evgeni Nabokov down on the ice and out of position, but the puck struck Islanders defenseman Calvin de Haan and ricocheted right to Backstrom, who scored with 7:15 left.
De Haan was injured on the play, helped off the ice, but returned in the third period. He took two shifts in the third, then left.
A puck that was ping-ponging off players in the Islanders zone wound up on the stick of Joel Ward in the slot, and his shot through Nabokov's legs evened the score with 4:59 remaining.
"We scored two quick ones there, that was important for us, so we could get our confidence back up," Backstrom said.
The Capitals were outshot 17-8 in the first period playing less than 24 hours after a 2-1 loss at the New Jersey Devils.
"We definitely played better last night, a lot sloppier tonight," Washington coach Adam Oates said. "I think fatigue factor, not even a whole day's rest, and they had two days off. ... It was sloppy, but if anything I thought our third period was probably the best in terms of eliminating chances."
The Islanders took a 3-1 lead 9:55 into the second against Holtby, who was making his third start since March 6.
"I thought he was good early, obviously in the second a couple shaky ones, but then he was great in the third and great in the shootout," Oates said.
The Islanders scored 12:55 into the game when forward Casey Cizikas beat Capitals defenseman Tyson Strachan to Michael Grabner's soft dump in behind the net. As Cizikas skated behind the cage, he backhanded a pass in the opposite direction right to Cal Clutterbuck, who swiped it in for his 12th goal of the season.
Holtby stopped a shorthanded breakaway by Clutterbuck less than three minutes later.
"Your job as a goalie is to keep your team in the game when there's some chances," Holtby said. "I thought we did a fairly good job. It was very impressive that we came back from being down. It was good to see."
An errant shot by Washington resulted in its first goal and a tie game 2:36 into the period. Kuznetsov sent a pass cross ice to Marcus Johansson, whose shot missed the net but turned into a perfect pass back to Kuznetsov, who took advantage of a helpless Nabokov for his third goal of the season.
It was his 13th game since arriving from the Kontinental Hockey League on March 10.
"I still feel he's still learning so much in terms of the pace of our play, the kind of game we play, the physicalness of it," Oates said. "And I thought tonight the Islanders' skating might help get him to skate a little bit more."
New York scored twice in a 5:39 span. John Persson got the first goal of his NHL career upheld by video review at 4:16. His backhand into Holtby seemed to be kept out, but Holtby's movement allowed it to trickle over the goal line before the whistle was blown.
"I wasn't sure what was going on," said Persson, who played his fifth NHL game. "I just fired it and then I went around the net; I wasn't even sure what he was reviewing. Obviously, I didn't want to celebrate too early, but it was nice to see it go in."
Grabner, who returned from a concussion to play for the first time since March 10, took his last shift with 6:04 left in the first period.
Capitals defenseman Mike Green did not play the final two periods because of an upper-body injury that will be evaluated Monday. Washington next plays Tuesday at the St. Louis Blues.