Real will face the winners of Thursday's (Friday AEDT) clash between Barcelona and Osasuna in the final on Sunday.
The rivalry, the comeback and the stakes combined to make this arguably the best match of the Spanish season, with two fearless sides battling for 120 minutes in a breathtaking derby.
"It was spectacular, but we don't enjoy matches like this," Real manager Carlo Ancelotti told Movistar Plus.
"The wear and tear has been tremendous. Because we are two very strong teams who have given everything."
Atletico, who inflicted Real's only loss of the season in September, started the game on top with Brazilian winger Samuel Lino almost opening the scoring with a curling strike that was saved by goalkeeper Kepa Arrizabalaga.
However, there was nothing Kepa could do when Antoine Griezmann lifted his corner into the box and defender Mario Hermoso sent a towering header flying past the keeper to give his side the lead in the seventh minute.
Real fought back through an Antonio Rudiger header from a Luka Modric corner in the 20th minute and a Ferland Mendy close-range strike, before Griezmann worked his magic in the 37th minute with a brilliant long-range strike.
Atletico goalkeeper Jan Oblak made a stunning stop with his legs to deny a Rodrygo close-range strike and the scores were level at halftime.
Real Madrid targeted the right side of Atletico's defence and Vinicius Jr. took advantage with a quick free kick to Dani Carvajal whose thunderous strike was brilliantly saved by Oblak.
Atletico retook the lead in the 78th minute thanks to a calamitous attempted clearance from goalkeeper Kepa Arrizabalaga who punched the ball on to his team mate Rudiger and it ended up in the net.
Real equalised through Dani Cavajal who converted a rebound after Oblak made two fine saves from Vinicius and Jude Bellingham.
With a penalty shootout looming, Joselu's header bounced off Savic into Atletico's net five minutes from the end of extra time to deliver a hammer blow to Atletico.
In the last play of the match, Diaz finished off a counter- attack into an empty net after Oblak went up to Real's box in a desperate attempt to equalise from a corner.
"I told my players that I'm really proud of them, they played a great game and we were very close, but it was an unfortunate outcome...," Atletico manager Diego Simeone told Movistar Plus.
The city rivals will clash again next week as Atletico host Real in an elimination game in the Copa del Rey last 16. In three weeks time, they will square up again at the Santiago Bernabeu in a LaLiga match.
