Owen Tippett and Scott Laughton scored for the Flyers (23-28-11), who have lost four in a row (0-3-1) and eight of nine (1-6-2). Hart made 22 saves.
"I thought we forechecked way better, supported pucks way better," Laughton said. "Got back on our heels a little bit in the third, but it's what happens when you're leading. They've got some good players and they had a tipped goal [by Kreider] and they finish in overtime.
"[We have to] keep moving forward. Thought it was a way better game from our group, and [we] go from here."
The Rangers took a 1-0 lead at 2:51 of the first period when Mika Zibanejad one-timed a feed from Tarasenko on a power play.
Tippett tied it 1-1 with a power-play goal at 15:05. New York defenseman Adam Fox tried clearing the puck out of the left corner, but his cross-zone pass went to Tippett, who beat Shesterkin from the bottom of the right circle.
Tippett had four shots on goal and led all forwards with 27:40 of ice time.
"I think if he hits the net that he has maybe one or two more," Philadelphia coach John Tortorella said. "But you could sense ... he was free, he was playing. He was looking at me wanting to play more."
Laughton gave the Flyers a 2-1 lead at 11:16 of the second period. He jarred a puck loose behind the net and it rolled to Kevin Hayes. Laughton then took a return pass, skated from below the goal line into the slot and beat Shesterkin with a shot over the goalie's right shoulder.
Kreider tied it 2-2 at 10:33 of the third when he redirected Tarasenko's shot from just inside the blue line.