Pageau tied it 3-3 from the slot with 2:09 remaining. The goal was upheld after Flyers coach Alain Vigneault challenged for offside.
"I thought we just stuck with it," Pageau said. "We came back after the first, refocused, made sure we were coming back and just getting back to the simple things, work hard, and do all the simple plays and get back as a unit of five and attack as a unit of five. I thought we did a really good job in the second and in the third to put pressure on them, and that's why we came back."
Hayes gave Philadelphia a 1-0 lead at 1:57 of the first period, beating Varlamov to the short side with a shot from the left circle.
Hayes made it 2-0 at 9:43, and Couturier extended the lead to 3-0 at 15:09 when he cut through the New York zone and scored inside the left post.
It was Couturier's first goal in 11 games this postseason; he was third on the Flyers with 22 during the regular season.
"We know we have another level to our game, and we need to bring it," he said. "Tonight was a step forward. We were skating well, we established a good forecheck, created a lot of chances. Would be nice to capitalize on a few more, but at least we're getting some chances, some good looks and we can build off that."