Varlamov made four saves in the first period against the Flyers but was much busier in the second, making 15 saves, including a pad stop on a rebound attempt by Philadelphia forward Claude Giroux at 5:12 to protect a 1-0 lead.
"We know how important the first game [is] each series; it's important to win the first one and have a good start, and we did that again today," he said. "But it's going to be a long series. It's going to be a lot of games again. The series is not just about one game, it's about winning four. We have to move on from this game and get ready for the next one."
Teams that win Game 1 are 485-220 (68.7 percent) winning a best-of-7 NHL playoff series, including 7-1 in the first round this season. Teams with a 2-0 lead are 328-51 (86.5 percent), including 4-0 in the first round.
Varlamov hasn't faced more than 29 shots in any of his 10 games this postseason.
"Anytime you have your goaltending playing well, it gives your team confidence," New York coach Barry Trotz said Tuesday. "I think we've played pretty well in front of him, and he's made some really good stops at key times, especially yesterday.
"I didn't care for our game really last night. I thought the first period was real strong, and then after that I thought we were very average. We've got a lot of work to do and help him out. But he's made big saves at key times. That's the key in goaltending, I think. You look at some of the great goaltenders in the National Hockey League, it's not necessarily the numbers or anything like that, it's do you make the save at the key time that allows you to win."
Varlamov signed a four-year, $20 million contract ($5 million average annual value) with New York on July 1, 2019. He replaced Robin Lehner, who was a finalist for the Vezina Trophy, awarded to the goalie voted as the best in the NHL, last season. Lehner became an unrestricted free agent and signed a one-year contract with the Chicago Blackhawks; he ended up with the Vegas Golden Knights on Feb. 24 as part of a three-team trade that included the Toronto Maple Leafs.
Varlamov was 13-13 with a 2.57 GAA and a .915 save percentage in 26 NHL postseason games before the start of the Stanley Cup Qualifiers. This is the first time he's played in the playoffs since 2014 with the Colorado Avalanche, who lost a seven-game series to the Minnesota Wild in the first round.
"He's been huge right from Day One since he came to us," Islanders center Brock Nelson said. "Not that we expect it, but obviously he goes out there and performs every night and is a backbone for us, so it's nice knowing that we have him behind us. If there's a breakdown or an error, he can bail us out.
"We've had good goaltending now for a while, [but] lately he's been on it making big saves, timely saves. That can win you games at this time of year."
