Disappointment for Barca was compounded by the sight of a visibly upset Ansu Fati going off injured in the second half of normal time at San Mames.
Fati only returned earlier this month after two months out with a hamstring problem, having also come back in September following 10 months out with a knee injury.
Another spell on the sidelines for Fati would be a huge blow to Barca's hopes of making La Liga's top four. Pedri also asked to go off in extra-time.
Barca were hoping to join them in the last eight but instead it is Athletic who advance, joining Madrid, Real Sociedad, Mallorca, Rayo Vallecano, Cadiz, Valencia and Real Betis in Friday's draw for the last eight.
Barca won the Copa del Rey last season and defending their title was probably their best chance of silverware this term.
January 20, 2022
They are out of sight in La Liga and were beaten by Madrid in last week's Spanish Super Cup, with the Europa League now their only realistic remaining route to a trophy.
Only 102 seconds had passed when Iker Muniain sent San Mames apoplectic by collecting Nico Williams' cross at the back post, before turning and bending into the far corner.
But Barcelona were level in the 20th minute, and from a similar angle, as Torres shifted right, inside Oscar De Marcos, and whipped a stunning shot into the net.
Inaki Williams hit the bar with a long-range effort late on but the real drama was still to come. Athletic thought they had won it in the 86th minute when Inigo Martinez prodded in at the back post ahead of the sliding Gerard Pique.
January 20, 2022
Barcelona, though, found a way back in the 93rd, as Dani Alves bicycle-kicked Jordi Alba's cross back into the area and Pedri's hit on the stretch was too powerful for Julen Agirrezabala's right hand.
Barca celebrated like it was the winner but there was extra-time to come and injuries. Fati went off and then Pedri asked to depart.
Then in the 103rd minute, a sliding Alba stopped Nico Williams' cross with his right arm and after consulting the replay, Jose Munuera pointed to the spot. Ter Stegen dived right, Muniain shot left.
January 20, 2022