Justin Williams brought the puck down the right side and into the slot, and left the puck for Wesley, who snapped it past Antero Niittymaki for the winner. Rod Brind'Amour reached the 20-goal mark for the 12th time in his 18-year career, and Andrew Ladd added a goal for the Hurricanes.
The defending Stanley Cup champions blew a two-goal lead yet recovered to keep pace in the Eastern Conference playoff chase. Carolina entered in eighth place in the conference standings, one point ahead of Toronto and the New York Islanders.
Grahame, who backs up Cam Ward, started for the first time since Feb. 1 and made 33 saves in his first victory since beating Florida on Jan. 16.
Alexandre Picard had a goal and an assist, and Dmitry Afanasenkov also scored for Philadelphia, which has lost four of five and ranks last in the NHL with 40 points.
Niittymaki made 27 saves for the Flyers, who had scored at least four goals in each of their last four trips to Raleigh - including a 5-2 victory on Dec. 31.
The Hurricanes scored twice late in the first period to take the early lead. Brind'Amour put the Hurricanes up 1-0 with about seven minutes left when he skated into the left circle, took a pass from Bret Hedican and snapped a shot that beat Niittymaki high to his glove side. Ladd made it a two-goal game about five minutes later when he worked his way to the net, fought through a check and hammered a rebound past Niittymaki.
Afanasenkov pulled Philadelphia to 2-1 midway through the second when he skated into the left circle, hesitated and snapped a shot past Grahame's glove side.
Picard tied it 2-2 with four minutes left in the third period when he tapped the rebound of Mike Richards' shot past Grahame.
Notes: Williams had two assists to record points for the first time in five games. ... Philadelphia has won once since trading Peter Forsberg to Nashville on Feb. 15. ... Simon Gagne, who had nine goals and four assists in his previous seven games, was held scoreless. ... The Flyers are the only team in the NHL with fewer than 50 points. ... Hedican played his 300th game with the Hurricanes. ... Tempers flared in the second period when Philadelphia's Todd Fedoruk and Carolina's Tim Gleason traded punches after Fedoruk gave Ray Whitney a shove from behind.