An own goal from on-loan Milan defender Stefan Simic gave the Rome club the lead in the 24th minute before Filip Djordjevic doubled the home advantage just two minutes later.
Felipe Anderson's long-range pot shot deflected in10 minutes from time put the result beyond doubt for the 2013 winners.
Stefano Pioli's side face an away trip to Torino in the last 16.
In a closely fought game at the Mapei Stadium, hosts Sassuolo edged past Serie B opponents Pescara courtesy of a 1-0 scoreline.
The former Argentina striker made no mistake from the spot in the 17th minute after he had been fouled in the box by Marco Rossi.
Perugia's task was made much harder in the 53rd minute when Guillermo Giacomazzi received a second yellow card, leaving the Serie B side to battle on, ultimately in vain, for the remainder of the second half.
Verona travel to face Serie A's dominant side Juventus in the fifth round.
