Carolina rallied from a 2-0 deficit and produced the game-winner shorthanded.
The Hurricanes' Rod Brind'Amour was called for tripping with 1:22 remaining in regulation, and Carolina was close to killing off the penalty in overtime when Wesley scored from the slot.
Wesley's last previous goal had come Feb. 5, 2006.
Alexandre Picard tied it at 2-2 with four minutes left in the third period.
Philadelphia has lost three straight and four of its last five.
Swim Week Miami Beach 2014 [PHOTOS]