The Nerazzurri, who came into the game at the San Paolo on the back of two successive Serie A defeats, looked to have forced extra time thanks to an unusually disciplined defensive display.
However, with just seconds remaining, Gonzalo Higuain was allowed to pick up possession from a throw-in and the Argentine took full advantage, advancing forward before sending the home side into the semi-finals with a fine finish.
Mancini was lost for words by the naivity of his side's defending, telling Rai Sport: "I don't know what to say. I'm speechless. You cannot concede a goal like this.
"We played a good game against a great Napoli, but you certainly cannot win if you don't score goals.
"I don't remember the last time I lost three games in a row. I must have been only 10.
"But I repeat: losing like this hurts. We were fools. We were too soft in the decisive moments of the game.
"Gonzalo is a great and clever player, he quickly realised that the was a chance to score and he did so.
"But you cannot concede a goal like the one we conceded. We have committed the same type of errors that cost us the game against Sassuolo last Sunday."
Inter, who sit 13th in the Serie A standings, resume their league campaign with a home game against Palermo on Sunday.
