Asham one-timed a bullet from the high slot off a pass from Craig Adams 8:33 into the final period to put the Penguins ahead 2-1.
Dupuis made it 3-1 for Pittsburgh only 1 1/2 minutes later, poking in a goal during a scramble in front of the net.
Adams also scored and Brent Johnson stopped 30-of-32 shots as the Penguins improved to 16-7-4, best in the NHL.
Jamie McBain netted his first two goals of the year for the reeling Hurricanes, who fell to 0-3-0 under new Coach Kirk Muller.
Brian Boucher made 37 saves for Carolina.