Carrier scored, and Thompson made 36 saves for the Golden Knights (29-18-4), who were also playing the second of a back to back following a 4-1 loss at the New York Rangers on Friday.
Vegas is 0-2-2 in its past four games and 1-5-2 in its past eight.
"We played a really good hockey game," Golden Knights coach Bruce Cassidy said. "Left a point on the table, obviously, but not through a lack of effort. Competing the right way and … all the things you're looking for. Obviously, right now we're having trouble finishing some plays, but if we keep that kind of effort up to keep the puck out of our net, we're going to be fine in the long run."
Lee gave the Islanders a 1-0 lead at 3:52 of the second period. Kyle Palmieri fell down as he was cutting toward the net, and Lee shot the loose puck into the net past the glove of Thompson.
Varlamov then kept the Islanders in front at 5:50 when he made a highlight-reel save. With the Golden Knights skating in on a 4-on-3 rush during a power play, he reached back with his stick and denied Reilly Smith a shot into an open net.
"I got lucky," Varlamov said. "The whole net was open and it happens like once a year. You shoot there 100 times, probably you're going to score 99 out of 100. But I'm glad I made that save. I mean, it was unbelievable.
"It feels amazing, it feels unreal. It's like one of those saves you make once a year. I'm glad I made a save. It was important, like in the right moment."