Los Leones won what was an absolutely thrilling contest 3-2 having taken the lead on three separate occasions after Barcelona equalised twice.
Iker Muniain fired Athletic into the lead after just two minutes, with Ferran Torres clapping back in the 20th minute to level things up and register his first goal for Barcelona since joining from Manchester City.
Inigo Martinez then thought that he had won it for Athletic in the 86th minute when he made it 2-1, only for Pedri to equalise again for Barcelona deep in stoppage time.
Then, in the first half of extra time, Muniain scored his second of the evening from the spot to make it 3-2 to Athletic.
The Basque side will face either Mallorca, Cadiz, Rayo Vallecano, Valencia, Real Betis, Real Sociedad or Real Madrid in the quarter final.