Holders Lazio, who are on a club-record run of 10 straight wins in Serie A and well in the title discussion, made short work of Cremonese in their easy 4-0 win in Rome.
The hosts were ahead in less than 10 minutes in front of a sparse crowd at the Stadio Olimpico thanks to Patric, who tapped home the opener after Cremonese stopper Michael Agazzi palmed the ball to him in the box, and the tie was done and dusted 16 minutes later when Marco Parolo bundled home the second from a Jony cross.
Simone Inzaghi made eight changes from the side that beat Napoli 1-0 at the weekend but kept Saturday's goalscorer Ciro Immobile, who took his tally to 23 in all competitions from the penalty spot 12 minutes after the break.
The impressive Jony turned provider again for the goal that rounded off the scoring, his fizzing cross from a set-piece being met with Bastos' bullet header in the final minute.
Troubled Napoli meanwhile are languishing in 11th in Serie A after winning just one league game since mid-October but eased into the quarters with a 2-0 win over Perugia thanks to two Lorenzo Insigne penalties.
"This needs to be a starting point. We need to stay calm and keep improving." Insigne told RAI Sport after the match.
Insigne struck the opening goal from the spot on 26 minutes after Mexico's Hirving Lozano went down inside the area under a challenge from Mardochee Nzita.
The Italy international added a second when Pietro Iemmello handled at a corner, with a penalty awarded following a VAR review.
Napoli goalkeeper David Ospina, whose blunder proved costly in their league loss at Lazio, then saved an Iemmello penalty before half-time following another VAR intervention.
Elseid Hysaj was harshly penalised for handling in the box after a cross from Diego Falcinelli hit the defender's leg before bouncing up against his trailing arm.
Inter Milan host Champions League-chasing Cagliari later on Tuesday while Serie A leaders Juventus host Udinese in the last 16 on Wednesday.