Dries Mertens put the visitors ahead after 27 minutes before goals by Lorenzo Insigne and Blerim Dzemaili in four second-half minutes sealed victory.
Rafael Benitez's side remain in third, two points behind Roma but 10 adrift of leaders Juventus.
It was the Verona team though who had the better of the early play.
Luca Toni blazed over from close range with Gokhan Inler also firing high over the bar from distance.
Gonzalo Higuain had Napoli's first real effort on goal when Rafael De Andrade punched his shot away but the visitors went ahead two minutes later through Mertens who curled home a shot inside the far post.
It heralded a period of domination by Napoli with Higuain firing inches wide and De Andrade saving well from Jose Callejon's shot.
Cabral then punched clear Toni's drive from the edge of the area but that was Hellas' last real effort on goal.
Goran Pandev fired over for Napoli with De Andrade saving Dzemaili's shot from the edge of the box in the corner.
However, Napoli then got the goal they deserved after 72 minutes when Christian Maggio produced some good play before sending a low cross into the box which found Insigne and he put the ball into the centre of the goal from close range.
The visitors got their third four minutes later after Dzemaili sent the ball into the bottom left-hand corner following a counter-attack.