Antonio Rudiger decided the pulsating tie with a header in the 115th minute, with Madrid to face Barcelona or Atletico Madrid in the final.
Real Sociedad, trailing 1-0 from the first leg, took the lead through Ander Barrenetxea, but Endrick’s fine chip pulled the 20-time winners level.
A David Alaba own goal and Mikel Oyarzabal’s deflected effort helped La Real open up a lead on the Spanish champions but Madrid battled back with goals from Jude Bellingham and Aurelien Tchouameni.
Oyarzabal grabbed his second in stoppage time to force extra-time but his side could not hold off Madrid until penalties, with Rudiger’s header the final word.
Ancelotti brought Vinicius Junior and Rodrygo Goes back into his starting lineup, leaving Kylian Mbappé on the bench to rest, with Endrick taking his place in an all-Brazilian attack.
The 18-year-old wonderkid striker who netted the only goal in the first leg, was active in the opening phases, sending a dangerous overhead kick narrowly wide of Álex Remiro’s far post.
Bellingham also threatened for the hosts but it was Real Sociedad who took the lead.
Barrenetxea broke in behind Lucas Vázquez to reach Pablo Marin’s flick-on and drilled a low finish home to put the Basques level in the tie.
Real Madrid soon found their equaliser. Vinicius played a sensational through ball down the left flank with the outside of his foot, sending Endrick in on goal.
The youngster, top scorer in the Copa del Rey, produced a gorgeous lobbed finish over Romero for his fifth in the competition.
Real Sociedad appealed for a penalty before the break when Takefusa Kubo went down in the box after being held by Vinicius, but the referee did not concur.
Ancelotti brought on Mbappé for Endrick, looking for a goal to shut the tie down for good. However, it was the visitors who were creating the better chances.
Madrid goalkeeper Andriy Lunin made a superb save to deny Martin Zubimendi and Oyarzabal wastefully fired the rebound wide.
Real Sociedad moved ahead on the night when Alaba deflected Marin’s cross through Lunin’s legs and into his own net after 72 minutes.
The Austrian defender, who returned from a long-term injury in January, was also involved in Real Sociedad’s third, deflecting Oyarzabal’s shot past Lunin.
It sparked Madrid in to action, with Vinicius turning his marker brilliantly, driving forward and crossing for Bellingham to strike in the 82nd minute.
Four minutes later and Los Blancos pulled level on the night with Tchouameni’s header badly fumbled by Remiro into his own net.
Just when it seemed Real Sociedad were out they managed to force extra-time, with Oyarzabal heading home Sergio Gómez’s free-kick in the 93rd minute.
Remiro saved well from Bellingham before the end of the regulation 90 minutes, with the pace slower, inevitably, as the game restarted.
Real Sociedad’s Jon Olagasti was booked for hacking down Vinicius as he threatened to escape on the left, with the visitors working hard to hold off the European champions.
With five minutes remaining Rudiger sent Madrid through to the final with a fine header from Arda Guler’s corner past the helpless Remiro.
On Wednesday Atletico Madrid host Barcelona in the second leg of the other semi-final, with the teams level at 4-4 on aggregate. (AFP)
