Adam Boqvist and Kyle Palmieri scored for the Islanders (24-20-7), who have won seven straight. Ilya Sorokin made 28 saves and Simon Holmstrom had two assists.
Boqvist was playing his first game for the Islanders after being claimed off waivers on Friday from the Florida Panthers.
"Obviously it's a lot in my head, I got in at 1:30 in the morning," Boqvist said. "Crazy day, but when you step on that ice you don't want to think too much, you just want to go out, win your battles and play for the team."
Jake Guentzel and Nikita Kucherov scored for the Lightning (27-20-4), who are 1-1-1 in the first three games of a five-game homestand. Vasilevskiy made 33 saves.
Guentzelgave Tampa Bay a 1-0 lead at 5:39 of the second period when he scored on a redirect of an initial shot by Erik Cernak.
"We got a point, obviously it's huge to get that at the end, but I don't think we had our best," said Guentzel, who will represent the United States in the 4 Nations Face-Off from Feb. 12-20. "Sometimes that's going to happen. You can't get frustrated, keep putting pucks on it and hopefully they'll go in."