Substitute Zouhair Feddal netted the dramatic winner for the visitors in stoppage time after 120 minutes, after Davide Moscardelli's late equaliser for the hosts had cancelled out Francesco Valiani's opener to force extra time.
The other top-flight team in action on Tuesday, Parma, enjoyed a much more comfortable passage, thrashing second-tier Varese 4-1 thanks to goals from Jaime Valdes, Gianni Munari, Raffaele Palladino and Aleandro Rosi.
Loris Damonte pulled a goal back in the 88th minute, but it was too little, too late to affect the outcome.
Elsewhere, Spezia overcame Serie B rivals Pescara 3-0 - the home team's task made all the easier by the first-half dismissals of Pescara pair Dario Zuparic and Aniello Cutolo.
Osariemen Ebagua set the ball rolling with an eighth-minute opener, and the visitors' hopes of getting back into the game received a big setback with the two red cards within half an hour.
Marco Sansovini then doubled Spezia's advantage on the stroke of half-time and substitute Emanuel Rivas sealed matters two minutes from time.
Finally, Serie B high-flyers Avellino ousted Frosinone - the lowest-ranked team left in the competition - 2-1.
Angelo D'Angelo and substitute Luigi Castaldo netted the goals, with Daniel Ciofani's last-minute strike scant consolation for the Lega Pro1B side.