Rafael Benitez's side have six points and seven goals from their first two games, with four of those goals coming from the Slovakian Hamsik.
The impressive Alberto Paloschi twice levelled for the hosts after Hamsik, with a superb long-range strike in the 14th minute, and then Jose Callejon, with a 27th-minute tap-in, put the visitors ahead.
An eventful first half also saw both sides create a host of chances, with Chievo striker Paloschi and his Napoli counterpart Gonzalo Higuain hitting the post.
However, Napoli seized control in the second period and Hamsik's close-range volley after a defensive mix-up gave them a deserved lead after 64 minutes.
Argentina international Higuain sealed the three points six minutes later when he took Lorenzo Insigne's deft through-ball and squeezed home his first Serie A goal in a one-on-one with Chievo keeper Christian Puggioni.