The hosts, boasting a number of changes in a bid to keep players fresh for their Serie A title hopes, were held at bay by the opposition.
Adam Ounas then came close with a stinging effort that beat the away stopper but bounced off the post and, come the half-time whistle, neither side had found a breakthrough.
Dries Mertens was the architect, slicing through the Zebrette defence with a diagonal through-ball to Insigne, who in turn bore down on goal and slotted past Scuffet at the far post.
The Partenopei will face the winner of Wednesday's last-16 clash between Atalanta BC and Sassuolo.
Napoli (4-3-3): Sepe; Maggio, Maksimovic, Koulibaly, Hysaj; Zielinski, Jorginho, Rog; Ounas (Allan 76'), Callejon, Giaccherini (Insigne 58')
Udinese (3-5-2): Scuffet; Nuytinck, Danilo, Bochniewicz; Stryger, Balic, Hallfredsson, Jankto (De Paul 46'), Pezzella (Widmer 70'); Matos (Lasagna 79'), Bajic