Mathew Barzal made it 3-0 at 13:31 of the first, when from the right circle he shot the puck around Rangers defenseman Tony DeAngelo and past Shesterkin.
Lee scored again with a redirection of a pass from Jordan Eberle for a power-play goal at 14:54 of the second period to make it 4-0.
Varlamov made 12 saves in the second.
"I thought we had a really good game today overall," Varlamov said. "We had a great start, scored a quick couple of goals and that set the tone. It's always easier to play like that, when you score first. Overall, I'm pretty happy with what we did today."
Rangers defenseman Jacob Trouba thought he scored at 9:15 of the second, but the goal was immediately waved off. The no-goal call stood after an official review.
The Islanders had a goal negated by a coach's challenge for goalie interference at 8:45 of the third period, when they outshot the Rangers 12-3.
"There's no statement game for us," Islanders coach Barry Trotz said. "It's the first game of the season against a rival. We know how important points are in a shortened season. We wanted to come out and play well. I thought we had a really good start.
"A couple minutes into the third period, about five minutes in, I thought we locked it down real good. I liked our game, but there's another level for us."
NOTES: It was the first shutout for an Islanders goalie in a season opener since Chico Resch against the Philadelphia Flyers on Oct. 7, 1976. … Rangers defenseman K'Andre Miller, the No. 22 pick in the 2018 NHL Draft, also made his NHL debut. He had one shot on goal and was minus-1 in 13:59.