Goals from Nico Gonzalez and Cristiano Biraghi were enough to see the Viola through
The Viola will face Club Brugge in the semi-finals. Vincenzo Italiano's side were beaten in the final of this competition last year by West Ham.
Italiano later said: "The penalty shootout was too dangerous, we had to force a little something and the choice paid off. Chino, Barak and Ikone came on very well, even Faraoni and Sottil if you enter with this mentality it's easier.
"We should have done it before but it seemed enchanted even tonight with the posts. We had prepared not to concede a goal in the first leg, we managed to play for everything tonight and we are very happy to be in the semi-final.
"This is for Joe Barone's wife and children. They were here and we are happy to have given them a smile, I know how sad these days are. We played for them too."