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Emmanuel Agyemang-Badu almost fired the hosts ahead inside the opening 10 minutes, but his powerful half-volley flashed just wide of the post.
Napoli's best chance of the first half came courtesy of Walter Gargano, who saw his curling effort from range come crashing off the post.
After the break, the visitors were denied by a fine double save from Orestis Karnezis, as the keeper got down low to keep out Gonzalo Higuain's angled drive, before he turned Jose Callejon's follow-up wide of the post.
Napoli struggled to come up with a response and Christian Maggio headed over the bar when well-placed from a corner.
It proved to be their last chance of the match as Udinese held on to claim the three points and heap further pressure on Rafael Benitez's side.