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.
Real Madrid will now face either Manchester City or Atletico Madrid in the last four.
