In Florence, midfielder Erick Pulgar converted a penalty after just nine minutes to the delight of the home crowd and new club owner Italian-American media magnate Rocco Commisso.
But Mertens pulled Napoli back after 38 minutes with Insigne adding a second from the spot just before the break.
Nikola Milenkovic's powerful header six minutes after the break kept Fiorentina, 16th last season, in the game.
Insigne set up Callejon for Napoli's third on 56 minutes and completed his own brace nine minutes later after substitute Kevin-Prince Boateng volleyed in a third for the hosts.
French midfielder Franck Ribery made his Serie A debut coming on the final quarter of an hour for Fiorentina and having a penalty appeal turned down.
"We conceded three goals, but we scored four away from home which isn't bad," said Napoli coach Carlo Ancelotti whose side play Juventus in Turin next weekend.
"Juve are moving from one coach to the next, but we'll try to play a great game with enthusiasm and focus, even if our most difficult match comes so early in the season."