Moise Kean's opener for Fiorentina was canceled out by Zsolt Nagy from the penalty spot. With scores ending in extra-time at 4-4 - and with the Viola having gone down to nine men - the tie went to penalties.
De Gea denied Puska Akademia captain captain Roland Szolnoki, as the visitors won the shootout 4-5.
Viola coach Raffaele Palladino later said: "It was essential to get through this round, I want to thank the lads because today they fought with heart, courage and great sacrifice. I'm sorry because we gave away the first half, the responsibility is mine while in the second half we were good, entering the field with a different mentality.
"Then there were the episodes of expulsions but our team spirit came out, now we must already think about the championship.
"I want to instill the mentality to dominate the game more and play better technically in the opponent's half, too many times we close ourselves off or don't find the free man. All things in which we are struggling but it's my job, it takes patience. The boys are committed, today they showed me that they are great men."