Dortmund opened the scoring when debutant Marco Reus struck early on.
Theodor Gebre Selassie levelled with 15 minutes left for Werder, but just over five minutes later Goetze pounced to steal the points for the champions.
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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.