Martin Necas and Jordan Staal each had a goal and an assist, and Antti Raanta made 26 saves for Carolina (5-2-1), which was coming off a 6-2 loss to the New York Islanders on Friday.
Wade Allison and Nicolas Deslauriers each had a goal and an assist, and Hart made 34 saves for Philadelphia (5-2-1).
"They should feel good about this point," Flyers coach John Tortorella said. "I know it [stinks] we lose it at the end. The comeback again from a 2-0 deficit and take the lead, some people kicked in that haven't kicked in for us. There was a lot of good things that happened there. So I think the club should feel good about itself."
Staal, who was playing in his 1,100th NHL game, gave the Hurricanes a 1-0 lead at 5:27 of the first period when he chipped the rebound of a shot off the end boards over Hart's left skate. He then helped set up the play that led to Jordan Martinook's goal at 15:04, which made it 2-0.
"You could tell he was relieved because it hasn't happened yet this year," Brind'Amour said of Staal's goal. "We know what he means to this group, and it's nice to see that happen."
Deslauriers cut it to 2-1 at 5:14 of the second period, scoring from the slot off a pass from Lukas Sedlak for his first goal with the Flyers. The forward signed a four-year contract with Philadelphia on July 13.
Owen Tippett tied it 2-2 with a power-play goal at 7:18. The forward has two points (one goal, one assist) in two games after missing five because of an upper-body injury.
"It's one of those things where you take it day by day," Tippett said. "Symptoms come and go, and you can't really rush anything."