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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The reputation of the Premier League means it is the most popular and one of the most watched leagues in the world.
Five English clubs will compete in the last sixteen of the Champions League for the first time ever as Liverpool, Manchester United, Manchester City, Chelsea and Tottenham face some difficult away matches.