Record 31-time winners Barca had the better of the clash at the Metropolitano stadium after the wild 4-4 first leg draw, and defeat leaves Diego Simeone's Atletico with little to play for in the rest of the season.
Real Madrid started brightly despite manager Carlo Ancelotti resting several key players and it took them 19 minutes to break the deadlock, with 18-year-old Brazilian Endrick providing a tidy finish from a counter-attack.
The French superstar did not train with his team-mates on Tuesday ahead of the game but coach Carlo Ancelotti said the striker would travel to San Sebastian.
Atletico Madrid manager Diego Simeone was delighted with his team's fighting spirit as they clawed back from two goals down to get a last-gasp 4-4 draw at Barcelona in an action-packed first leg of their Copa del Rey semi-final on Tuesday.
Ancelotti rested several key starters ahead of Saturday's LaLiga derby against Atletico Madrid and next-week's Champions League playoff clash at Manchester City.
Record 31-time Copa del Rey winners Barcelona will face Valencia in the quarter-finals, after they were paired in Monday's draw.
Real Madrid midfielder Eduardo Camavinga has been sidelined with a hamstring injury suffered during the club's 5-2 Copa del Rey win over Celta Viga, the club said Friday.
Record 31-time Copa del Rey winners Barcelona were drawn at home against Real Betis in the last 16 on Wednesday.
"(Luka) is a gift for football and even more for those who can enjoy it up close," Madrid coach Carlo Ancelotti said. "The fans, the players, the coach... for me it's a great gift. The way he prepares for this kind of match, as if it was a final. It's a very important example for our youngsters.
Players boarded a barge known as the 'Gabarra' and sailed 13 kilometres down the river from the affluent town of Getxo to the Bilbao city hall, where the party continued.
Athletic have been through years of heartache, having lost their previous six consecutive Spanish cup finals and a Europa League final to Atletico Madrid in 2012.
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
