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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Since Guus Hiddink arrived at the helm, Chelsea have improved considerably after a tricky start to the campaign.
Newcastle United are staring down the barrel of relegation to the Championship for the second time in the past eight years.