Philadelphia is home for its final two games, against the New Jersey Devils on Saturday and the Capitals on Tuesday.
"We win and we get to fight another day," Flyers coach John Tortorella said. "We're going to have to do the same thing when we play Saturday. We're going to have to be ready to go and see if we can extend this."
Artemi Panarin scored his 47th goal and Jonathan Quick made 23 saves for the Rangers, who have lost two in a row in regulation for the first time since Jan. 18-20.
"The whole game wasn't good enough," Rangers center Mika Zibanejad said. "We knew they were going to be desperate. We didn't match their intensity or desperation level. We don't execute what we're supposed to do. We're slow to do things. Just overall not good enough."
The Rangers (53-23-4) have a three-point lead on the idle Carolina Hurricanes for first place in the Metropolitan Division. New York has two games remaining, both at home, against the New York Islanders on Saturday and Ottawa Senators on Monday. Carolina has three games to play.
"We're fighting for the division right now," defenseman Ryan Lindgren said. "Carolina is right on our tail. There's no time for us to let up. We've got two games left. We want to play good hockey and be ready for the playoffs obviously."
York gave the Flyers a 1-0 lead, scoring from the left circle at 4:21 of the first period. The Flyers scored first twice in their eight-game skid.
Konecny pulled up with the puck on the far side, causing K'Andre Miller to lose an edge and go down. He found York coming into the zone, setting him up for the shot off the rush.
"It just gives you that energy," Tortorella said. "I thought we lost ourselves a little bit there when we took the penalties and they scored, but we kept our patience."
Panarin tied it 1-1 with a 5-on-3 power-play goal from the left circle at 18:19.
Brink made it 2-1 at 3:29 of the second period, when his attempted pass to Poehling, who was cutting across the front of the net from left to right, went in off Miller's stick.
"The harder you work the luckier you get," Cates said. "That's what I believe and what just happens in this weird game. It was nice to get some bounces, but you can't rely on them, obviously."