It was Varlamov's first game against the Avalanche since signing a four-year contract with New York on July 1, 2019. He played the previous eight seasons with Colorado.
"Of course I'm excited to meet my previous team," said Varlamov, whose shutout was his second this season. "I played for them for a long time, for eight years. Of course this one is a special one.
"It was really weird, I'm not going to lie. It was very weird. But I guess I just tried to stay focused on my own things. I just tried to do my own job. I'm glad we won."
WATCH: [All Avalanche vs. Islanders highlights]
Anders Lee scored for New York (26-12-3), which was 0-3-1 in its previous four home games, all at Nassau Coliseum.
Pavel Francouz made 32 saves for Colorado (25-14-4), which had won two straight.
"That's the staple of their game, real committed defensively, real good in almost all defensive aspects of the game," Avalanche coach Jared Bednar said. "That was the type of game we were expecting.
"We had some chances to capitalize and we didn't. I thought [Varlamov] was really good in their net, [Francouz] was good in our net. They controlled the play in the first. We controlled the play in the second. It's tough to create scoring chances and when you had them, you just needed someone to capitalize on one. They did it once and we weren't able to."
Lee scored with a wrist shot from the left face-off circle at 6:54 of the third period to give New York a 1-0 lead. It was the Islanders first goal in 139:21.