Carolina took a 2-0 lead but needed the goal by Tlusty to avoid overtime for what would have been the 14th time this season.
This game was originally scheduled for Tuesday, but foul weather forced a 24-hour postponement.
Alexander Semin scored in the first period and Nathan Gerbe tallied in the second to put the Hurricanes in front. Philadelphia came back thanks to a backhander from Claude Giroux with 35 seconds to go in the middle period and a power-play goal from Scott Hartnell early in the third.
The win went to Anton Khudobin, who made 21 saves. Steve Mason stopped 29-of-32 shots for the Flyers, who lost their second in a row.
Celebrity Families of 2014 [PHOTOS]