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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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.