Despite their European success in recent years, this was only Real’s second domestic cup semi-final in the past 11 years.

Defending a one-goal lead from the first leg, the hosts found themselves behind around the quarter-hour mark when Ander Barrenetxea slotted through Andriy Lunin’s legs following Pablo Marin’s through ball.

That opener came somewhat against the run of play, with Endrick twice going close for Real at the other end – including a speculative overhead kick – while Vinicius Junior saw an effort saved by Alex Remiro.

The hosts, though, eventually had their equaliser in the 30th minute, when the two aforementioned Brazilians linked up in style.

Vinicius’ stunning pass with the outside of his boot released Endrick, who delicately lofted over Remiro.

After the interval, Carlo Ancelotti’s side looked to add to their lead, although clear-cut opportunities were rare. Spectators had to wait until the 60th minute for the next shot on target, when Jude Bellingham stung Remiro’s palms from distance.

Nevertheless, it was instead Sociedad who took the lead, albeit courtesy of a huge slice of fortune.

Dogged work from Marin got him to the byline and he fired a low cross into the box more out of hope than expectation, but it found the heel of David Alaba, who could do nothing as the ball trickled between Lunin’s legs and into the back of the net.

Remarkably, Los Txuri-urdin were soon in front on aggregate, and again, Lady Luck was on their side.

Again, Alaba had his head in his hands as Mikel Oyarzabal’s drive hit him before striking the post and nestling in the back of the net.

Madrid, though, do not know when they are beaten, and they reduced the deficit within seconds when Bellingham side-footed into the net from the middle of the penalty area following some brilliant wing play from Vinicius.

Before long, Los Blancos were back in control and moved ahead once more in the tie, with Aurelien Tchouameni heading home, although Remiro will feel he should have done better.

With seven minutes of injury time added, the game appeared set up for another twist, and it didn’t disappoint.

A free-kick was perfectly delivered by substitute Sergio Gomez, allowing Oyarzabal to head beyond Lunin and send the encounter into extra-time, although only after Remiro had denied Bellingham.

At the start of the additional 30 minutes, the home team exerted the initial pressure, but Remiro was not forced into action.

Just as it seemed the game was meandering towards a penalty shoot-out, Real struck the killer blow to their opponents. Antonio Rudiger was the goalscorer, heading home from a corner with five minutes remaining.

Unlike previously, Sociedad were unable to find a response as they exited the competition at this stage for the second time in as many years.

They will now turn their attention to the race for a European spot via LaLiga, while Real will watch Wednesday’s other semi-final second leg between Atlético Madrid and Barcelona to see who they play in the showpiece.

Atletico took the lead with Alvaro Morata scoring from close range following Nahuel Molina’s low cross.

Real were 11 minutes from going out but equalised when Rodrygo beat three players and scored a wonderful goal.

Atletico’s Stefan Savic was sent off for two bookable offences before Karim Benzema put Real ahead in the 104th minute with Vinicius Jr adding a third.

Benzema steered a low shot past Jan Oblak after the visitors had failed to deal with Marco Asensio’s low cross and Vinicius Jr sealed the victory as he passed the ball into the bottom corner after a fine run.

Barcelona and Osasuna beat Real Sociedad and Sevilla respectively in their quarter-finals on Wednesday and Athletic Bilbao won 3-1 at Valencia on Thursday to join them in the semi-finals.

The draw for the final four will take place on Monday, 30 January with the first legs in early February and the second legs in the beginning of March, before the final on 6 May.

Earlier on Thursday, La Liga said it “strongly condemns acts of hatred and intimidation against Vinicius Jr” after an effigy of the Real Madrid winger was hung from a bridge near the club’s training ground.

A banner reading “Madrid hates Real” was attached to the bridge’s railings, with Atletico calling the incident “repugnant”.

Brazil international Vinicius Jr, 22, said last month the Spanish league had to act on racists attending games after a minority of Atletico fans sang racist songs aimed at the player before Real’s 2-1 win in September.

The visitors, who had won their previous nine games, had Brais Mendez sent off in the first half for a dangerous tackle on Sergio Busquets.

Dembele settled the tie when he ran onto Jules Kounde’s ball down the right wing before hammering in from an angle.

Gavi almost put the game beyond doubt but hit the bar.

Osasuna also made it to the last four of the competition after beating Sevilla 2-1 in extra time.

Ezequiel Avila looked to have won it in normal time for the hosts only for Youssef En-Nesyri to level in the 94th minute.

On-loan Barcelona winger Abde Ezzalzouli scored in the first half of extra time to send Osasuna through.