With the score 0-0 from the first leg, Dortmund went 1-0 down and then 2-1 behind at home.
But two goals in injury time, from Marco Reus and Felipe Santana, secured a dramatic and famous win that puts them in the last four.
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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.