The Ghana forward played on Monday for his country but made it home to help Athletic vanquish the Spanish champions and pile more pressure on their coach Xavi Hernandez.
Athletic struck inside the first minute through Gorka Guruzeta, but Barcelona turned the game around with goals from Robert Lewandowski and 16-year-old starlet Lamine Yamal.
Oihan Sancet netted early in the second half to take the game to extra-time and then teed up Inaki Williams to put the Basques ahead.
His brother Nico Williams added the finishing touch late on to complete a superb night for the hosts.
After Barcelona's recent humbling by Real Madrid in the Spanish Super Cup final Xavi admitted the team needs to win a trophy this season.
Eight points off the top in La Liga, the record 31-time Copa del Rey winners saw the cup as an accessible route to silverware.
In their way were former Barca coach Ernesto Valverde and his Athletic side, 23-time cup champions, behind only the Catalans.
Roared on at a raucous San Mames, Athletic took the lead after just 36 seconds.
With Barcelona's defence disorganised and Athletic benefiting from a couple of lucky breaks, the ball sat up kindly for Guruzeta and he turned it home.
Conceding early goals has been a big problem for Barcelona this season - remarkably this was only the third fastest.
Xavi's side have shown good grit to fight back in matches though and they did it again after defender Alejandro Balde went off with a hamstring injury.
Lewandowski charged down Yuri Berchiche's clearance and deflected it into the net after 26 minutes, and six minutes later the visitors were ahead.
The electric Yamal cut in from the right and flashed a shot into the bottom corner from the edge of the box.
Athletic goalkeeper Julen Agirrezabala tipped Ferran Torres' strike narrowly wide before half-time and his team pulled level early in the second half.
Sancet subtly peeled off behind Frenkie de Jong to guide a header home past the helpless Inaki Pena.
Guruzeta should have put Los Leones ahead but headed off-target, with Inaki Williams falling to the floor in frustration on the sidelines.
The Ghana international, who raced back from the Africa Cup of Nations to participate, soon entered the fray.
His brother, Nico Williams, the architect of Sancet's leveller, bent a shot narrowly past the post as the match lurched back and forth, while Yamal dinked wide when through on goal.
The youngster should have won the tie for Barcelona with five minutes to go, racing through again and rounding the goalkeeper but screwing his shot wide.
In extra-time Athletic struck just before the interval, with Inaki Williams tapping home after his first effort hit the post and bounced back to him.
Pena made a brave stop to deny Inaki Williams from close range before Nico Williams killed the game off with a sublime finish after his brother knocked the ball into his path.
In the other quarter-final of the night, Mallorca scored three first-half goals before losing a man to beat LaLiga leaders Girona 3-2.
