Dortmund had opened their campaign in style last week but the defeat is a blow to their hopes of defending the title.
Bayern scored a late winner to beat Wolfsburg after losing at home to Borussia Moenchengladbach last weekend.
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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.