In a game in which the two teams managed just two strikes on goal between them, the hosts took the lead after 23 minutes when 17-year-old Udinese defender Matteo Palma beat Jonathan David to a Weston McKennie cross only to turn the ball into his own goal.
Palma, a Berlin-born Italian youth international, fouled McKennie in the 67th minute to concede a penalty that Manuel Locatelli tucked away.
The match opened the Italian Cup round of 16.
Napoli and Inter Milan have home ties on Wednesday. On Thursday, cup holders Bologna host Parma and Serie A leaders Milan visit Lazio.
The final tie, between Roma and Torino, will be played on January 13.