Romelu Lukaku put Inter ahead after just 22 seconds from kickoff. Borja Valero doubled their advantage later in the half before Lukaku added his second after halftime.
The Rossoblu netted a consolation later in the second half through Christian Oliva, but Andrea Ranocchia stamped out any comeback with the Nerazzurri's fourth before fulltime.
Napoli made light work of Serie B side A.C. Perugia with an easy 2-0 victory at the Stadio San Paolo in Naples.
The home side advanced to the quarterfinals thanks to a first-half double from Lorenzo Insigne, who got on target with penalties in the 26th and 43rd minute.
Lazio's hot run of form in all competitions continued after thumping US Cremonese 4-0 at the Stadio Olimpico in Rome.
Patric hit the opener earlier on before Marco Parolo added a second later on.
The lead was extended with a second-half penalty from Ciro Immobile and a late strike from Bastos compounding the visitors' misery.