Alex Ovechkin did not score and remains at 730 NHL goals. His next will tie Marcel Dionne for fifth in history.
The Capitals captain did not play in overtime after he pulled up in the neutral zone and went to the bench with 2:49 left in the third period.
"I don't even know what is wrong," Laviolette said. "I know that he wasn't available during that time."
Samsonov made 15 saves in the first period, including a breakaway against Mathew Barzal at 18:34.
New York outshot Washington 15-4 in the period, but the Capitals held a 14-7 edge in the second period.
"If we could have gotten a lead, I think it would have given us a little spark hopefully for the second," Islanders coach Barry Trotz said. "This is a good team we're playing and it's going to come in waves. … That was a battle."
Varlamov made a pad save on Carl Hagelin's redirection at 3:08 of the second.
Josh Bailey nearly gave the Islanders the lead at 12:02, but Samsonov managed to get his blocker on a shot from the left circle.
Samsonov then stretched out his left pad to deny Kyle Palmieri during an Islanders power play at 10:42 of the third period, and Barzal flipped a shot over the crossbar on a breakaway at 15:45.
"I thought he made big saves the entire night, but most of them were in the first period," Laviolette said.
The Islanders were held to two shots on goal on two power plays in the third period.
"They were playing aggressive," New York defenseman Nick Leddy said. "They were moving their feet. I think we got set up a couple of times. We just didn't happen to get a whole lot to the net. We had a couple of chances, especially 'Palms' there. He had a nice look and it almost went in."
Jordan Eberle had the best chance to score in overtime, but his shot from the left circle was stopped by Samsonov's pad at 3:51.
Anthony Beauvillier had the lone goal for the Islanders in the shootout in the second round, and the Capitals responded with back-to-back goals from Backstrom and Kuznetsov.
"I thought both teams played really well tonight," Beauvillier said. "There was not a lot of space. Obviously, 'Varly' was really good and unfortunately we couldn't get the win for him."
NOTES: It was the second straight 1-0 game between the teams; Varlamov made 29 saves in a New York win here on April 6. … Varlamov is tied with Philipp Grubauer of the Colorado Avalanche and Marc-Andre Fleury of the Vegas Golden Knights for the NHL lead in shutouts. … Washington forward Anthony Mantha had his five-game goal streak end. He scored in each of his first four games since being acquired in a trade with the Detroit Red Wings on April 12. … Forward Michael Raffl made his Capitals debut after being acquired in a trade with the Philadelphia Flyers on April 12. Raffl, who had been out with an upper-body injury, had two shots and two hits in 13:32 of ice time. … Islanders forward Cal Clutterbuck had one shot in 13:02 after missing three games with an undisclosed injury. Islanders forward Travis Zajac was a healthy scratch.