Greene made it 1-0 with a slap shot from the blue line at 6:06. It was the defensemen's first playoff goal since April 16, 2010, against the Flyers as a member of the New Jersey Devils.
He also led the Islanders with four blocked shots in 18:37 of ice time, including with 1:31 left in the first period, shooting his right leg out on the rush to get his skate blade on a Travis Konecny one-timer from in close.
"The goal, got to the middle, I saw a shot lane, put that through," Greene said. "The other one, I can't let that pass get through, and then once it did then it's desperation mode and I got lucky."
Greene was acquired in a trade with the Devils on Feb. 16 and played 10 games before the season was paused. He entered the lineup in Game 2 against the Florida Panthers in the Stanley Cup Qualifiers.
"Greene is solid back there, such a veteran presence," Lee said. "He's been around, he's seen a lot, played in a ton of big games. His experience has been extremely valuable for us and our [defense] corps. ... And tonight to get us going, that was huge."
After making four saves in the first period, Varlamov made 25 in the final two, including 15 in the second.
"I feel like I haven't done anything really special so far," Varlamov said. "I just play the game. It happens now that I get back-to-back shutouts, but it's amazing the play in front of me. But I've already moved on from this game and looking forward to playing the next one."