Nolan Patrick scored two goals in the third period for Philadelphia, which rallied from 4-1 in the third period but gave up the go-ahead goal 28 seconds after tying the game.
The Flyers (40-26-14), who hold the second wild card into the playoffs from the Eastern Conference by four points over the Florida Panthers, could have clinched a postseason berth with a win if the Panthers lost to the Nashville Predators in any fashion (Florida won 2-1). Philadelphia also could have clinched with one point and a Florida loss in regulation.
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"We played all year trying to be in the playoffs, and now we have a good chance of making it," Flyers captain Claude Giroux said. "But we've got to earn it first."
Mathew Barzal had two goals and an assist for the Islanders (33-37-10), to reach 82 points as a rookie.
Barzal put New York ahead 5-4 at 13:32 of the third period when Anthony Beauvillier sent him on a breakaway for Barzal's second goal of the game.
"I think in a month or so, when I'm looking back at the whole season, just look at the whole deal, first year in the NHL, I think it'll be pretty special," Barzal said. "Right now, I've got a lot on the table still. It's not really something you can dwell on during the season."
Patrick scored two power-play goals in 4:09 in the third period to make it 4-3. He scored his first goal at 4:07 when he dove to poke a rebound past goalie Thomas Greiss (27 saves), then scored again when he redirected Jakub Voracek's feed at 8:16.
Giroux tied it 4-4 at 13:04 with a one-timer from the right face-off circle for his 30th goal.