Carlo Ancelotti was full of praise for his players and the Real Madrid fans after another "inexplicable" Champions League semi-final victory sent the 14-time winners through to the final.
With the tie poised 2-2 after last week's first leg in Munich, Bayern put themselves in control when Alphonso Davies scored a stunning goal in the 68th minute, but Madrid produced a sensational comeback with Joselu netting in the 88th and 91st minutes for a 4-3 aggregate win.
Spanish champions Real Madrid face another German side, Borussia Dortmund, in the Wembley final on June 1, after they defeated Paris Saint-Germain in the other semi-final.
The last time Real Madrid won the Champions League, in 2021/22, their run to the title was filled with great escapes. They fought back from a 2-0 aggregate deficit against PSG in the last 16, needed extra time to get past Chelsea in the quarter-finals, then scored two late goals to force extra time in their semi-final with Manchester City.
So it was no surprise to see the competition's most successful team mount a late comeback against Bayern on Wednesday night. Real will now aim to complete the double after clinching La Liga last weekend.
“It's happened again. Something that has happened time after time, which is inexplicable, has happened again," Real manager Ancelotti said. "It's magical. We played well, it was a tough game but we had control, we didn't lose our heads when they scored and we got back into it.
"I'd like to thank the fans, the club and above all, the players, they've delivered a season nobody expected, myself included. I didn't imagine that this squad would be capable of what they've achieved."
Madrid had enjoyed the better of the match but it seemed Davies' superb strike had set up a repeat of the 2013 all-German final in London.
Instead, Joselu consigned Bayern to a first trophyless season since 2012, his first goal coming after a handling error from Bayern goalkeeper Manuel Neuer, who had superbly kept Madrid at bay until that point.
"There were world-class saves after world-class saves, but that can happen, it's football," said Bayern defender Matthijs de Ligt.
Neuer produced a superb double save to stop Madrid nosing in front, tipping the electric Vinicius Junior's effort on to the post and then saving Rodrygo's follow-up.
Bayern's Serge Gnabry limped off injured with Davies replacing him before England captain Harry Kane's first flash of danger, but Andriy Lunin pushed his vicious volley around the post.
Neuer palmed away a Vinicius free-kick as the teams ended a high-intensity, relentlessly frantic first half level.
The second period was just as fraught, with Davies' cross deflecting onto the roof of Lunin's goal.
Vinicius was unplayable down Madrid's left and Rodrygo prodded his low cross agonisingly wide of the far post.
Neuer, 38, made superb saves to deny both Brazilian forwards again before the hour mark as Madrid ratcheted up the pressure.
However, just as in Munich, when Los Blancos were at their strongest, Bayern pounced. Canada international Davies cut inside from the left and smashed a shot beyond Lunin into the far top corner after 68 minutes.
Madrid thought they had levelled moments later through Nacho but after a VAR review, the strike was ruled out as the defender had grabbed Joshua Kimmich's face before shooting.
Tuchel took off Kane, still chasing the first trophy of his career, in the final stages to try and hold on to victory but it slipped through his fingers as Madrid produced more indelible European memories.
Neuer, who had been sublime to this point, fumbled what appeared a simple Vinicius strike and substitute Joselu gratefully bundled home from close range.
With Bayern reeling Madrid struck again, decisively. It was Joselu once more, the striker on loan from second-division side Espanyol, turning home after Antonio Rudiger cut the ball back to him.
The goal was ruled offside but officials overturned the decision after review, and after nearly 15 minutes of stoppage time, Madrid earned the chance to go for their 15th Champions League trophy.
"There have been a lot of times we have looked dead and buried, but we have that mentality of never say die," said Real midfielder Jude Bellingham.
Bayern were left raging at full time after De Ligt netted but play had been stopped beforehand for an offside flag against Noussair Mazraoui.
"I think it's pretty clear and there is no doubt that is against the rules of modern football," said Bayern manager Thomas Tuchel, who will leave the club imminently.
"A disaster. An absolute disaster. With Real's second goal, they let the play continue. The clear rule is that the scene must continue. The first mistake was made by the linesman, the second by the referee."
