Victory means that the Partenopei have the highest points tally ever for a runner-up in the Italian top flight, finishing four behind champions Juventus.
First-half goals from Arkadiusz Milik and Jose Callejon also proved enough to condemn Crotone to relegation, as they were unable to pick up the result needed to overhaul Cagliari, Udinese and SPAL on the final day, despite a late response from Marco Tumminello.
Napoli edged in front with a quarter of the match played, with Lorenzo Insigne picking out Milik to head the ball into the top corner.
Insigne also created the second goal, this time providing a pass for Callejon to volley home from close range at the back post.
A good save from Alex Cordaz then prevented Callejon from making it three, before thwarting Milik to give his side just a glimmer of hope of rescuing themselves.
The closest the hosts came to a third was through Dries Mertens's effort 74 minutes in, which Cordaz managed to parry onto the post.
Crotone found a way through in added time through a Tumminello curler, placing the ball out of Pepe Reina's reach to deny the Spaniard a clean sheet on his Napoli farewell.
There was to be no equaliser, however, meaning relegation for the Pitagorici after two years in the top division.
NAPOLI (4-3-3): Reina; Hysaj, Albiol, Koulibaly, Mario Rui; Allan (Rog 77'), Jorginho, Zielinski; Callejon, Milik (Mertens 67'), Insigne (Hamsik 79')
CROTONE (4-3-3): Cordaz; Sampirisi, Ceccherini, Capuano, Martella; Barberis, Mandragora, Rohden (Stoian 46'); Trotta (Tumminello 64'), Simy, Nalini (Ricci 58')