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Riccardo Montolivo sought to test the Napoli keeper early on, but his shot from distance flew just a few inches too high of the visitors' crossbar.
The away side took the lead with their first real chance of the game when Insigne was able to pick out Allan inside the area, leaving the midfielder to fire comfortably beyond Diego Lopez in the 13th minute.
The home stopper was called into action not too long after too, and he had to produce a strong save down low to prevent Gonzalo Higuain from doubling Napoli's lead.
Napoli came out of the traps quickest in the second half, and they doubled their advantage through Insigne, who latched on to a through-ball from Higuain to finish beyond Lopez.
The travelling side made it three not too long after in the 67th minute, when Insigne returned to bag a brace with a memorable free kick that evaded the Milan wall and nestled into the top corner.
AC's misery was compounded with just 13 minutes remaining, as Rodrigo Ely somehow managed to turn the ball into his own goal to help seal an emphatic victory for Napoli on the road.