Bilbao are the second most successful team in the history of the Copa del Rey, their 23 titles behind only Barcelona's 31, their most recent coming in 2020-21.
Sergi Darder fired the decisive spot-kick as Real Mallorca beat Real Sociedad 5-4 on penalties to book their place in the Copa del Rey final on Tuesday.
Inaki Williams raced back from Africa to face Barcelona in the Copa del Rey and sent Athletic Bilbao into the semi-finals by scoring a crucial goal in their 4-2 extra-time triumph on Wednesday.
Defeat snapped Madrid's 21-match unbeaten run.
"It's a positive result, our play was positive in many moments, and we're in the last 16," Barca coach Xavi told Movistar.
Atletico Madrid progressed with a 3-1 win at third-tier Lugo after Memphis Depay struck twice, while Real Betis were knocked out in a 1-0 defeat by Alaves.
"This Copa arrived at an important moment, we came through very tricky ties, it's a deserved prize against strong opponents," added the coach
Carlo Ancelotti's Madrid earned their 20th Spanish Cup in their 40th final, ending a nine-year wait to lift the trophy again
Real Madrid coach Carlo Ancelotti thinks the Copa del Rey final against Osasuna on Saturday could be his last -- although he has thought the same in the past and been wrong, he explained Friday.
Turkey international Arda Turan scored a hat-trick as holders Barcelona thrashed third-division side Hercules 7-0 in the second leg of their Copa del Rey last-32 tie.
Turkey international Arda Turan scored a hat-trick as holders Barcelona thrashed third-division side Hercules 7-0 in the second leg of their Copa del Rey last-32 tie.
A much-weakened Real Madrid went goal crazy in the Copa del Rey on Wednesday, smashing third-division minnows Cultural Leonesa 7-1 away from home in the first leg of their last-32 encounter.
Neymar will stay with Barcelona after signing a five-year contract, the club have announced.
Barcelona striker Luis Suarez is a doubt for next month's Copa America after hobbling off in tears midway through the Copa del Rey final win over Sevilla Sunday with a muscle injury.
A Spanish judge on Friday overturns a controversial ban on Barcelona fans flying Catalan separatist flags at Sunday's Copa del Rey final against Sevilla in Madrid.
Luis Enrique has lauded Lionel Messi as Barcelona's leader and "the best wherever he plays".
Barcelona forward Lionel Messi is out of hospital and recovering after a non-surgical procedure to cure his kidney problems.
Barcelona have confirmed that Lionel Messi is undergoing further tests on kidney problems which first emerged in December but expect the forward to return to training on Wednesday.
First-half goals from Munir El Haddadi and Neymar give Barcelona a slender lead in their Copa del Rey quarter-final tie against Athletic Bilbao with a 2-1 first-leg win.