Serie A Wrap: Napoli and Milan both held to draws
Goran Pandev put Rafael Benitez's ahead with two goals in three first-half minutes but they were unable to hold onto their lead.
A Federico Fernandez own goal and Bruno Fernandes' first Udinese goal brought the visitors level with 20 minutes to play.
Just two minutes later, Blerim Dzemaili put Napoli in front again as he stole in to turn home after Zeljko Brkic spilled Gonzalo Higuain's effort back into danger.
But Francesco Guidolin's side were not to be beaten and they secured a point thanks to Dusan Basta's 80th-minute leveller from close range.
In Saturday's other game Milan were grateful to striker Mario Balotelli as he produced a stunning 84th-minute free-kick – his second goal of the game – to secure a 2-2 draw away at Livorno.
The point means Milan are still yet to win back-to-back games this season and ensured the pressure continues to grow on head coach Massimiliano Allegri
The visitors had taken an early lead through Balotelli but goals from Luca Siligardi and Paulinho had put Livorno on course for a much-needed victory.
However, the 23-year-old striker ended that dream with a 30-yard strike that Francesco Bardi could get nowhere near in the Livorno goal.