Oskar Lindblom, Sean Couturier and Travis Konecny scored for Philadelphia (37-37-2), which failed to qualify for the postseason for the second time in three seasons.
"We dug ourselves a big hole at the beginning of the year," Couturier said. "We pushed hard and battled hard to get back in the (playoff) race, we were close at one point. We lost some crucial games, and at the end of the year, it was disappointing the way we finished. ... It feels like guys kind of had a little letdown at the end."
Carter Hart made 24 saves for the Flyers, who ended the season with five straight losses.
Staal scored from the left slot to put the Hurricanes ahead 4-3 at 7:33 of the second period. Dougie Hamilton's pass from behind the net bounced off Couturier's skate to Staal.
"I'll take it," Staal said. "Don't know exactly where that went but heck of a pass from Dougie. Fortunately, it ended up around my skates and fortunate to get it in."