Pittsburgh's Evgeni Malkin scored to open the tiebreaker, but Jussi Jokinen of Carolina quickly answered with a goal of his own. After Fleury stopped Jeff Skinner in the second round, Neal fired a shot past Hurricanes goalie Cam Ward.
Fleury made Neal's tally hold up with a dramatic toe save on Eric Staal to win the game and hand the Penguins a third straight victory.
The Penguins goalie finished with 25 saves and Malkin got Pittsburgh's regulation goal.
Ward took the hard-luck loss despite turning aside 40-of-41 shots.
Jamie McBain scored for the Hurricanes, who suffered their fourth straight loss in Pittsburgh.