The visitors took the lead when a through ball from Mario Rui set up Insigne’s lob shot that Victor Osimhen pounded in.
Koulibaly added another after the break with a blistering effort following another free kick. Subsitute Hirving Lozano then sealed it with a curled shot six minutes from time to conclude a counterattack.
With its fourth consecutive win to open the season, Napoli moved two points clear of defending champion Inter Milan and AC Milan.
Udinese, which entered the game unbeaten, was left five points behind Napoli.
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