Napoli fans sensed it was going to be a long night when Dries Martens blasted a penalty against the crossbar after only four minutes of the round-of-sixteen tie.
The hosts, playing their first game of the competition after receiving a bye along with the other top eight Serie A sides, struck the woodwork for a second time in the 35th minute when Walter Gargano's effort was deflected onto the inside of post.
It got worse as Cyril Thereau put Udinese ahead just before the hour when he put the finishing touch to a 50 metre run by Nabil Jaadi, who unselfishly rolled the ball to him.
Napoli, who face Inter Milan in the quarter-finals, were awarded another penalty in the 65th minute after Duvan Zapata's shirt was tugged and this time Jorginho stepped up and stroked the ball home.
Udinese's Silvan Widmer was sent off for a second bookable offence three minutes later, but they still forced the match into extra time.
Napoli regained the lead when Marek Hamsik scored with an unstoppable shot from the edge of the area.
Not to be outdone, Panagiotis Kone levelled for the visitors with an equally stylish goal, hooking in a low shot from the edge of the area after the ball was played in from the right.
The first eight penalties in the shootout were converted before Allan's effort for Udinese was saved by Mariano Andujar, leaving Gonzalo Higuain to convert the final kick and send Napoli through.
In the other quarter-finals, AC Milan will host Lazio, Parma play Juventus and AS Roma meet Fiorentina.
(Writing by Brian Homewood in Berne. Editing by Patrick Johnston)
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