Xherdan Shaqiri salvaged a point for the league leaders by scoring the equalizer in the 59th minute, cancelling out Thorben Marx's 21st minute penalty.
Marx scored with Borussia's first shot on goal after Jerome Boateng was adjudged to have blocked Tolga Cigerci's cross with his arm.
Bayern had 26 shots on goal compared to for the visitors' six, but struggled to create clear chances in the face of dogged defending and an impressive performance from Borussia goalkeeper Marc Andre ter Stegen.
The result leaves Bayern 12 points clear of Bayer Leverkusen ahead of the rest of the final games before the winter break.
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