An incredible first period saw four goals scored, three of which came from defenders, with Mario Hermoso sending Atletico ahead in just the sixth minute, before Antonio Rudiger and Ferland Mendy both netted in the space of nine minutes to make it 2-1 to Real Madrid.
Antoine Griezmann levelled the scores in the 37th minute, before a 78th-minute own goal from Rudiger sent Atletico 3-2 ahead, but Real Madrid were back on terms at 3-3 in the 85th minute through Dani Carvajal.
Extra-time was therefore required to separate the two sides, and Carlo Ancelotti's side scored a fourth in the 116th minute when Stefan Savic turned a Carvajal cross into the back of his own net, before Brahim Diaz added a fifth in the final moments, with the current La Liga leaders advancing to Sunday's final, where they will meet either Barcelona or Osasuna.
Real Madrid goalkeeper Kepa Arrizabalaga made a smart save to keep out a strike from Samuel Lino in the sixth minute, but Atletico made the breakthrough from the resulting corner, with Hermoso heading a delivery from Griezmann into the bottom corner.
Los Blancos had a shout for a penalty in the 14th minute when Rodrygo went down under a challenge from Saul Niguez, but the referee was not interested.
Ancelotti's side managed to level the scores in the 20th minute, though, with Rudiger heading a corner from Luka Modric into the bottom corner of the net.
Real Madrid came close to a second in the 27th minute through Rodrygo, but the Brazil international's strike was excellently kept out by Atletico goalkeeper Jan Oblak.
Both sides continued to threaten, but it was Real Madrid that scored the game's third goal in the 29th minute, with Mendy flicking a low cross from Dani Carvajal into the back of the net.
The response from Atletico was strong, and it became 2-2 in the 37th minute, with the first non-defender on the scoresheet, as Griezmann brilliantly managed to work space on the edge of the penalty box before picking out the corner with a wonderful strike.
Rodrygo came close to netting towards the end of the first half when he fired towards goal from inside the six-yard box, but Oblak was on hand to make an excellent save.
The first chance of the second half came for Lino in the 54th minute, but the Brazilian's effort from inside the box just missed the target.
The match settled in the period that followed, which was not a shock considering the hectic first half, but Carvajal should have sent Real Madrid ahead in the 71st minute.
The defender had time and space in a central position inside the box, but Oblak was once again on hand to make a smart save, with the two teams level at 2-2 heading towards the final 15 minutes.
Atletico managed to score the game's fifth goal in the 79th minute, though, with Rudiger turning into the back of his own net under pressure from Alvaro Morata.
Kepa came to punch a dangerous cross into his box, but it rebounded off Rudiger and into the back of the net, with Simeone's side taking the lead in bizarre circumstances.
Real Madrid were not finished, though, and it became 3-3 in the 85th minute through Carvajal, who scored with a powerful strike after a period of pinball inside the Atletico box, with Vinicius Junior and Jude Bellingham somehow both being denied before Carvajal registered.
Brahim Diaz had a chance to win it for Los Blancos in stoppage time, but he missed the target with his strike after dancing his way into the Atletico box, and the first semi-final of the Spanish Super Cup, which certainly produced in terms of entertainment, entered extra-time.
As is often the case, there was little of note in the opening 15 minutes of the additional period, but Real Madrid made it 4-3 in the 116th minute when Savic turned a Carvajal cross into the back of his own net under pressure from Joselu, with the ball bouncing up and over Oblak, before Diaz scored a fifth in the second minute of added time at the end of the contest, rolling into an empty net after showing incredible pace to get there, with Atletico's goalkeeper up the field for a corner.
Ancelotti's side will now wait to learn whether it will be Barcelona or Osasuna in Sunday's final.