Thomas Mueller set them on their way early on before Mario Gomez struck twice.
Martin Harnik got two goals himself as Stuttgart threatened a fightback, but Bayern held on to seal a remarkable clean sweep of league, Champions League and cup.
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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.