Bologna did pull a goal back from the spot but they couldn't overcome an impressive Napoli side...
The Partenopei took an early lead thanks to a Hassan Yebda header, before Ezequiel Lavezzi made it two midway through the first half, but Bologna gave themselves a lifeline when Riccardo Meggiorini scored from the spot in the second half but they couldn't find another meaning Napoli go through to the last eight.
Both sides took the chance to rest some key players with Edinson Cavani and Marco Di Vaio only making the bench for their respective teams.
Napoli started much better and it took them only nine minutes to find the back of the net after Yebda glanced in a header from a corner.
The Partenopei looked in the mood and found acres of room on the right hand side with youngster Juan Armada having a nightmare. Lavezzi and Marek Hamsik took turns tormenting the left-back, who failed to cope time after time.
Dominating their opponents, Napoli made it two on 23 minutes after Hamsik once more found plenty of room on the right side, before putting in a low cross which was dummied by Juan Zuniga for Lavezzi, who showed sublime close control before poking past a helpless Cristiano Lupatelli.
Bologna were impotent as an attacking force before Napoli went two goals ahead, but they did manage to emerge from their own half as the home team took their foot off the gas a little bit.
However, Napoli were never put under any real pressure as Bologna's only shot on target came from Meggiorini and that was a relatively tame effort which Gennaro Iezzo easily gathered as the Partenopei went into the break firmly in control.
The second half was only minutes old before it was stopped due to a problem with the floodlights, but after referee Andrea Romeo consulted with the players and coaches the game restarted.
Emerging from the break with purpose, the home team were dominant once more but shot themselves in the foot when Emilson Cribari stuck a leg out and brought down Meggiorini in the penalty area. The forward was looking for the foul and Cribari obliged with a needless challenge.
The forward picked himself up and gave Bologna a way back into the tie.
However, the away side could find an equaliser and Napoli were never in any trouble keeping the ball and eventually saw the game out and as result are through to the last eight of the competition where they will play Inter.