Ngone struck in the seventh minute with a left-foot drive that bounced in front of Salvatore Sirigu and squirmed under his body into the net, leaving the keeper shaking his head in disbelief.
Sirigu was at fault again five minutes later when Ngonge lashed another left-footer from a difficult angle, the ball flying across the face of the goal before nestling in the side netting to make it 2-0.
Sitting second in Serie A and with Palermo 12th in the division below them, Napoli were too big, too strong and too quick for their visitors and Juan Jesus nodded home a corner to make it 3-0 just before the break.
Palermo were reduced to 10 men when Aljosa Vasic was sent off, receiving a straight red card for dangerous play in the 59th minute to end any slim hopes Palermo had of a comeback.
David Neres slotted home the fourth in the 70th minute, pressing to help win the ball back before slotting it coolly into the net.
Midfielder Scott McTominay came off the bench and scored with his first touch, rifling home a pass from fellow substitute Romelu Lukaku in the 77th minute to complete the rout.