Rodrygo forced extra time and Karim Benzema struck the decisive blow after Chelsea had taken a three-goal lead.
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Real Madrid reached the UEFA Champions League semi-finals with a remarkable 5-4 aggregate success after extra time against Chelsea, who were on the verge of one the competition's greatest turnarounds when leading 3-0 late in normal time.
15': Mount reduces aggregate deficit for Chelsea
51': Rüdiger's thumping header levels tie at 3-3
75': Werner completes turnaround for Blues
80': Rodrygo volleys in wonderful Modrić cross
96': Benzema header edges Madrid through
Match in brief: Madrid progress after all-time classic
Thomas Tuchel switched to a back four with his Chelsea side trailing 3-1 from their home leg and the tactical tweak had the desired effect as the visitors dominated the first half and halved the aggregate deficit when Mason Mount whipped a 15th-minute finish past Thibaut Courtois. Madrid showed little of their first-leg swagger in the opening 45 minutes, Karim Benzema's deflected drive the only moment of significance for Carlo Ancelotti's charges.
Chelsea maintained their momentum after the restart and stunned the Estadio Santiago Bernabéu again when Antonio Rüdiger towered above the home defence to head in Mount's corner and level the aggregate score. Then, after Marcos Alonso's crashing drive had been ruled out for a handball and Benzema clipped the bar at the other end, the turnaround looked to be complete as Timo Werner tiptoed through and drove his finish under Courtois.
As in the last round against Paris, Madrid appeared beaten only to conjure something special and haul themselves back into the contest. Luka Modrić was the architect, producing a stunning outside-of-the-boot cross for substitute Rodrygo to volley in. Yet even that would not have been enough to force extra time had Chelsea substitute Christian Pulišić not spurned two late chances.
The home side were buoyed by their second chance and Benzema would not be denied a starring role. The French forward, who headed in twice in the first leg, repeated the trick from Vinícius Júnior's teasing cross to give the hosts the lead again overall and this time Tuchel's men could not respond, Kai Havertz and Jorginho passing up late openings.
PlayStation® Player of the Match: Luka Modrić (Real Madrid)
"Heavily influenced the control of midfield. Was the springboard to start attacks and cleaned up in a defensive role. The assist was simply sensational."
UEFA Technical Observer panel
Simon Hart, Real Madrid reporter
When it comes to this competition, Real Madrid simply have the knowhow. They were on the ropes at times against a Chelsea side who outplayed them for long stretches, posting 29 attempts to Madrid's ten, yet they found a way. Again two of their old stagers, Modrić, with that wonder ball to Rodrygo, and Benzema, with his fourth goal of the tie, made decisive contributions. Their reward is a 31st semi-final appearance.
Joe Terry, Chelsea reporter
Tuchel spoke about being perfect for 90 minutes in this game and the brutal fact is that Chelsea weren't: two lapses totalling about 60 seconds have cost them. And when you're against Madrid in the Champions League, and they're sitting on a 3-1 lead from the first leg, that's all they need. It will be hard to take for the players but they should take immense heart for their fearlessness, quality and determination to fight against the odds.
Nacho, Real Madrid defender: "Incredible – one more of these nights here at the Bernabéu. The tie became very difficult for us. It was similar to the Paris game. It gives you goosebumps each time you experience a night like this at the Bernabéu. We have a lot to improve, especially from the start of the game, but that's Real Madrid – we are taught from a young age that you never give up."
Luka Modrić, Real Madrid midfielder: "Unbelievable. We were dead until the goal that we scored. I cannot say that we played a bad game but they took their chances well and scored the goals. We knew even after the first game it would be tough because for me they're the most difficult team to play against."
Antonio Rüdiger, Chelsea defender: "The positive is that we didn't give up. Not many teams can come here and dominate them, as we did. Over the two legs, if you make the type of mistakes that we did, you get punished. We controlled the game and didn't panic. I think we did it quite well. In the second half, we scored goals and did everything we could."
Thomas Tuchel, Chelsea coach: "We are very disappointed and very proud at the same time because we played a fantastic match. We deserved what we had – we scored four goals and we had big chances to score even more. We were unlucky, that is why we are disappointed. We got beaten by pure individual qualities."
Jorge Valdano, Movistar
"Madrid were dominated by a great team for most of the match but survived and got through. There is something miraculous about it. Things happen here that are hard to believe."
- Madrid have scored in their last 28 home matches in the Champions League knockout phase. The last time they failed to was against Barcelona in the 2011 semi-finals.
- The Merengues had not conceded more than one goal in any of their last seven Champions League matches before this game.
- Prior to this season, the most goals Benzema had scored in a Champions League campaign was seven. He has now scored seven in his last three games in the competition.
- Chelsea have now scored in their last 11 away matches in the quarter-finals of the Champions League.
- The Blues are the first English team to score three goals at the Estadio Santiago Bernabéu since Manchester United in 1968.
Timo Werner – 10
Mason Mount – 10
Luka Modrić – 9
Antonio Rüdiger – 9
Real Madrid: Courtois; Carvajal, Nacho (Lucas Vázquez 88), Alaba, Mendy (Marcelo 78); Modrić, Casemiro (Rodrygo 78), Kroos (Camavinga 73); Valverde, Benzema, Vinícius Júnior (Ceballos 115)
Chelsea: Mendy; James, Thiago Silva, Rüdiger, Alonso; Kanté (Ziyech 100), Loftus-Cheek (Saúl Ñíguez 106), Kovačic (Jorginho 106); Mount, Havertz, Werner (Pulišić 83)
Real Madrid will face Manchester City in the semi-finals. The first leg will take place in Manchester on 26 April, with the second leg at Estadio Santiago Bernabéu on 4 May.