The top-seeded Flyers had fought back from a 3-1 series deficit in the NHL's quarantine hub in Toronto, forcing game seven with back-to-back overtime wins.
But the Islanders controlled the decisive contest to reach the conference final for the first time since 1993.
They'll open against the Tampa Bay Lightning in Edmonton on Monday, with a chance to reach the Stanley Cup final for the first time since 1984.
Greiss replaced Semyon Varlamov, who had started 14 of the Islanders' first 15 post-season games, and produced the first playoff shutout of his career.
Varlamov allowed nine goals in the previous two games, prompting Islanders coach Barry Trotz to opt for Greiss.
"I wanted the freshness," Trotz said. "I also knew that with 'Greisser,' just his demeanor, that he'd be a good choice for us. We've been blessed with two goalies. You try to ride the hot guy as long as you can, and if you have to go to the next guy, you do that."
Mayfield gave the Islanders a 1-0 lead at 9:27 of the first period with a rising shot that made it under the crossbar.
"Game-winner in Game 7 to go to the Eastern Conference Final, I don't think anyone thought I'd be scoring it," Mayfield said. "But I had a lane and just happy I could put it away."
Defenseman Greene made it 2-0 at 13:13 with a goal from the left circle. Derick Brassard set things up, holding the puck on the right until he spied Green in the opposite circle.
Nelson scored to make it 3-0 at 11:26 of the second period, finishing a two-on-one off a feed from Josh Bailey.
"Everybody wants to score goals and get on the attack," Nelson said. "Thought we had a pretty balanced score sheet up and down the roster. Contributions from the (defensemen) today, every line getting in on it, guys stepping up and making plays at the right time."
Bailey added another assist on Beauvillier's empty-net goal at 13:12 of the third period after Flyers coach Alain Vigneault pulled Carter Hart -- who made 22 saves -- for an extra skater.
New York held the Flyers to 16 shots, killing off both of Philadelphia's power-play opportunities to take their penalty kill to 13-for-13 in the series.
The last time the Islanders reached the conference final they fell in five games to the eventual Stanley Cup champion Montreal Canadiens.