The Blues travelled to the Santiago Bernabeu for their quarter-final second leg needing a minor miracle, having been beaten 3-1 at Stamford Bridge a week prior.
Thomas Tuchel’s side came racing out of the blocks and took the lead on the evening just 15 minutes into the game.
Mason Mount burst through and produced a brilliant finish to make it 3-2 to Real Madrid on aggregate.
Parity in the tie was restored five minutes after half-time.
Mount swung in a corner and Antonio Rudiger rose highest to head home.
Chelsea thought they had taken the lead in the tie when Marcos Alonso rifled the ball into the top corner from close range.
However, VAR ruled out the goal for a handball by Alonso moments prior to finding the net.
Chelsea were livid with the decision but they composed themselves and, with 15 minutes later, they took the lead for the first time in the tie.
Timo Werner was ice-cold as he showed incredible composure to find the back of the net.
But Real came roaring back and made it 4-4 on aggregate just five minutes later.
Luka Modric produced a stunning ball and Rodrygo volleyed home from 10 yards out.
Christian Pulisic came close to winning it with virtually the last kick of normal time, but his effort went narrowly over the bar.
The game went to extra-time and Karim Benzema restored Real’s lead in the 96th minute.
Benzema, who had hit the bar earlier in the evening, made no mistake as he headed home Vinicius Jr’s cut-back.
Chelsea had two late chances through Kai Havertz and Jorginho but both of their efforts went agonisingly wide.
The Blues couldn’t find the goal they needed and so it was Real who progressed to the semi-finals.
They will play either Manchester City or Atletico Madrid in the last four.
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