Staal's power-play goal at 11:49 of the third period gave the Hurricanes a 3-2 lead, capping a comeback after they trailed 2-0, but Konecny's goal forced overtime.
"We've been in that position before and not been able to tie it up," Flyers captain Claude Giroux said. "But to be able to get that goal to tie it up was a relief."
Carolina, which started a four-game road trip, has points in seven straight games (5-0-2) and is six points behind the Maple Leafs with 12 games to play.
"Obviously it would have been nice to have two points here," Lindholm said. "I think the other games we work our best to get the two points, and today we didn't. It would have been nice today."
The Flyers took a 1-0 lead on Provorov's goal with 57.8 seconds remaining in the first period. Weise scored at 11:03 of the second.
Skinner banged in the carom of Noah Hanifin's shot off the end boards at 12:37 of the second and Lindholm one-timed a feed from Phillip Di Giuseppe past Mason at 18:56 to tie the game 2-2.
The goal extended Lindholm's NHL career-best point streak to seven games.
After Staal put the Hurricanes ahead, the Flyers were able to swing the momentum in the final minute of the third and take two important points on Schenn's overtime goal.
"We needed a bounce back from the last game [a 6-2 loss to the New Jersey Devils on Thursday], and that's what we did tonight," Provorov said. "We'll go on the road trip with a good energy."