With the top four sides all playing teams in the bottom half in this round, second-placed Frankfurt let a one-goal lead slip to draw 2-2 at home with Augsburg.
Bayern dominated proceedings but were unable to turn pressure into points, with Harry Kane's absence telling.
The German giants were one goal up at half-time after Dayot Upamecano headed from a corner 18 minutes in.
The France defender turned villain early into the second period when he passed directly to Mathias Honsak, who gleefully accepted the invitation to equalise.
Bayern manager Vincent Kompany, whose side were eliminated in the German Cup by Bayer Leverkusen midweek, brought Musiala off the bench and the 21-year-old blasted in five minutes later.
Leon Goretzka's late goal put Bayern on course but Heidenheim scored again, this time through Niklas Dorsch, to keep the heat on the home side,
Musiala would have the final say, dribbling on the counter from the half-way line and scoring his second.
Bayern veteran Thomas Mueller praised Musiala, telling Sky "what he's doing, we expect that from him now.
"We're happy we've got him."
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