Marco Reus put Dortmund ahead but Bayern rallied through Arjen Robben to level.
A Daniel van Buyten own goal and Ilkay Gundogan put Dortmund ahead again, before Robben scored another. Reus sealed victory for Dortmund to give Pep Guardiola a losing start to his competitive career as Bayern coach.
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Chelsea have struggled to build momentum in the opening stages of the campaign and are currently looking over their shoulders at the relegation zone.
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