Lucas Evangelista and Cyril Thereau struck the decisive goals after the match had finished 2-2 at the end of 90 minutes, securing a place in the fifth round.
Having fallen behind to a fourth-minute goal via Milan Djuric, Udinese mounted a comeback and by half-time they were ahead at the Stadio Comunale Friuli.
Midfielder Allan levelled the scores after 20 minutes, before Bruno Fernandes put Andrea Stramaccioni's side ahead before the break.
Cesena brought themselves back into the game five minutes before the hour as Davide Succi powered a header past Simone Scuffet.
However, they were reduced to 10 men six minutes later as Massimo Volta was dismissed for a poorly timed challenge on Fernandes.
Ultimately that cost Cesena their place in the next round as Udinese used their numerical advantage to dominate extra-time, with Lucas and Thereau sealing the win.
Empoli are also into the next round after they produced an impressive home display to beat Genoa 2-0.
Diego Laxalt opened the scoring on four minutes as he met a Manuel Pucciarelli pass to fire home, but they had to wait until the 72nd minute to secure the win.
It was well worth the wait, though, as Levan Mchedlidze produced an acrobatic back-heel flick beyond goalkeeper Eugenio Lamanna from the edge of the penalty area, letting the ball go over his head before applying the finish.
Meanwhile, a Richmond Boakye double gave Atalanta a 2-0 win over Serie B side Avellino, the Juventus loanee striking twice in six first-half minutes to set up a meeting with Fiorentina in the next round.
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