The Polish forward scored early on to put the German side ahead but Cristiano Ronaldo had levelled by half time.
But Lewandowski pounced on a loose ball in the box twice to make it 3-1 before netting a penalty to wrap up an emphatic victory.
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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.