Real rallied to score twice late on but could not find the all important third goal that would have knocked out Dortmund.
Instead the German side progress 4-3 on aggregate, where they will face compatriots Bayern Munich if they avoid defeat to Barcelona.
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Since Guus Hiddink arrived at the helm, Chelsea have improved considerably after a tricky start to the campaign.
Newcastle United are staring down the barrel of relegation to the Championship for the second time in the past eight years.