Real Madrid will now face either Manchester City or local rivals Atletico Madrid in the last four.
Despite the fightback put in by Chelsea, Real Madrid booked their place in the Champions League semi-finals following a 3-2 defeat after extra time in the second leg at the Santiago Bernabeu on Tuesday. The La Liga table-toppers go through the top four 5-4 on aggregate.
Chelsea had looked on course to complete an astonishing comeback after taking a 3-0 lead in their quarter-final second leg with goals from Mason Mount, Antonio Rudiger and an excellent individual effort from Timo Werner.
Yet the hosts came roaring back, with Rodrygo finishing from a superb pass by Luka Modric to level the tie 4-4 on aggregate and force extra time when Karim Benzema headed home to decide a remarkable encounter.
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Real Madrid will now face either Manchester City or local rivals Atletico Madrid in the last four, with the English side taking a 1-0 lead into their quarter-final second leg in Spain on Wednesday.
It was a Champions League night for the ages at a packed stadium as a brilliant Chelsea came within 10 minutes of completing a rescue act that their manager Thomas Tuchel had said they were unlikely to accomplish.
His team took the game to Real Madrid as they looked to turn around a two-goal deficit from the first leg and it took only 15 minutes for Mount to open the scoring from a through ball behind the defence.
Chelsea was even more aggressive after the break and scored their second in the 51st minute when Rudiger guided home a header from a Werner corner to level the tie 3-3 on aggregate.
Four minutes later, Werner found Marcos Alonso inside the area and he thrashed his finish into the top corner only to see his effort chalked off by the video assistant referee who had spotted a handball from the Spanish left-back.
With the visitors on top, however, it was only a matter of time until they scored their third which arrived in the 75th minute from a brilliant piece of individual skill from Werner, who left two opponents on the ground before dispatching a precise shot past Courtois and inside the far post.
In the very next move, Kai Havertz was denied by former Chelsea keeper Courtois, who made a magnificent save to stop his close-range header.
Yet just when Real looked completely out of the game, Modric worked his magic, finding substitute Rodrygo at the far post with a delightful pass with the outside of his boot that the Brazilian volleyed past Edouard Mendy in the 80th to take the game to extra time.
That was the cue for Vinicius Jr and Benzema to show up and rescue Real Madrid from what would have been one of the biggest shocks they had ever suffered at home.
Vinicius worked his way inside from the left corner and sent a lovely cross onto Benzema's head, and the Frenchman bulleted it past Mendy as the ecstatic home fans exploded with joy.
"It was another incredible night, what a sweet-tasting loss," Modric told Movistar Plus.
"We didn't give up until the end. Despite going 0-3, I don't think we played a bad game. Another night of suffering, but the important thing is that we have qualified."
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