Vanecek kept it a one-goal game just over a minute later with a save against Nelson after a turnover, followed by a sliding pad save on Mathew Barzal with 4:20 left.
New York outshot Washington 20-10 in the third period.
"Usually most of those games we will win if they keep it down to one [goal], so I think he did everything we can ask from him today," Capitals forward Lars Eller said of Vanecek. "I think he played as well as we could ask for."
Varlamov made 12 saves in the second, then preserved the shutout with a save on Conor Sheary's wrist shot at 8:38 of the third period.
"That was a key save probably in the game," Varlamov said. "I'm glad I had a chance to stop that one. Then we turn around and score a goal after that."
The Islanders, who defeated the Capitals 8-4 at New York last Thursday, hit the post twice and a crossbar in the first two periods.
"The first, I think the edge probably went to them," Washington coach Peter Laviolette said. "I liked the way we played the second period. It seemed like we climbed back in it a little bit. It was tight. Not a lot of room."
NOTES: It was Varlamov's fourth shutout of the season. … The Islanders have a point in 18 of 19 home games (16-1-2). … New York general manager Lou Lamoriello tied Glen Sather (Edmonton Oilers, New York Rangers) for second in NHL history with his 1,319th win. Nashville Predators GM David Poile leads with 1,436. … Bailey has 160 goals and 340 assists in 904 games with the Islanders. …. New York has outscored the opposition 46-25 in the third period. … It was the first time this season the Capitals lost when they allowed two or fewer goals (14-1-0).