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The home fans marked the 25th anniversary of the Hillsborough disaster with a chorus of 'You'll Never Walk Alone' ahead of kickoff.
It looked like the visitors had taken a quick lead when Ivan Perisic found the back of the net less than three minutes into the tie, but the goal was ruled out for a foul on Dortmund keeper Roman Weidenfeller.
The hosts soon found their way into the match and broke the deadlock after 12 minutes when Mkhitaryan blasted the ball in from outside the box.
Marco Reus had a chance to double their lead a couple of minutes later, but a poor first touch prevented him from getting his shot away.
Perisic and Naldo had further chances for the visitors, but some solid defending from Jurgen Klopp's side prevented them from getting the goal back.
The misses proved costly as Lewandowski slotted in the second for Dortmund just before the break, curling it into the top corner from a Reus delivery.