Nelson gave New York a 1-0 lead at 6:52 of the second with a wrist shot from the right circle after a turnover by defenseman Philippe Myers.
Couturier tied it 1-1 when he redirected Justin Braun's shot at 15:19 of the second period.
Greiss kept the game tied with a sprawling glove save on Jakub Voracek's rebound attempt with nine seconds remaining in the second.
Philadelphia had the final 12 shots on goal in the second period and outshot New York 17-3.
"In my opinion, that was the worst game that we've played to this point in the playoffs," Islanders coach Barry Trotz said. "You talk about the ultimate goalie partner for anybody, Thomas Greiss is it. Whatever you want, he's all-in. He's practiced really well, he gives you everything he's got every day at practice, and then when you throw him in, he can do a lot of different things. Tonight, he was outstanding. The second period, he was unbelievable and in the third period, he was the same. You talk about key stops at key times, he made them."
Pageau put the Islanders in front when he was sprung on a breakaway by Scott Mayfield and got enough on a wrist shot that it got past Elliott and rolled over the goal line.
Pageau, who was acquired from the Ottawa Senators hours before the NHL Trade Deadline on Feb. 24, leads the Islanders with a plus-13 rating and is tied with forward Anthony Beauvillier for the team lead with seven goals this postseason.
Vancouver Canucks center Bo Horvat leads the NHL with nine goals.
"The payoff for getting him is obviously paying off right away," Nelson said of Pageau. "He's a big part of the team up the middle, provides a lot of depth, face-off guy, and he's scoring big goals. It's a guy you can lean on in any situation really. He plays in all three zones. He kinds of does it all, so he's a guy you appreciate and love to have on your team."
Nelson scored again when he finished a give-and-go with Josh Bailey at 11:12 of the third to make it 3-1.
Provorov scored on a long wrist shot with 1:05 remaining for the 3-2 final.
"It's tough," Elliott said. "You want to get a win, especially in this pivotal game here. But we're down, but we're not out and I think the guys realize that. We've just got to play our game.
"I thought everything they got that we kind of gave them. It was a little bit of unforced errors and I think that they're good at that, so we've just got to keep that out of their hands. I think if we play our consistent game, we'll give ourselves a good chance."
NOTES: Elliott made 18 saves in the first period. … New York has outscored Philadelphia 8-1 in the third period in the series; it outscored the Washington Capitals 7-1 in the third in the first round. … Flyers forward Oskar Lindblom took part in warmups but was scratched; he has not played since being diagnosed with Ewing's sarcoma, a form of bone cancer, in December. ... The Islanders lost the 1993 Wales Conference Final to the Montreal Canadiens in five games.