Bayern came from behind to lead after Manuel Neuer's mistake gifted Christopher Nkunku the opener, with substitute Jamal Musiala and Muller scoring in the space of four first-half minutes.
Justin Kluivert responded rapidly with his first Bundesliga goal to send the teams into half-time level, before Emil Forsberg's header looked set to send Leipzig to the summit.
However, Leipzig sat back and Muller connected with a cross from Kingsley Coman with 15 minutes remaining, with a point enough to ensure Bayern remained the league's pacesetter.
After Marcel Sabitzer struck the crossbar, Nkunku put Leipzig ahead when Neuer raced out of his box but failed to beat the Frenchman to Forsberg's throughball and left him with a vacant net to pass into.
Javi Martinez was forced off with a muscular injury but his replacement Musiala arrowed in an equaliser five minutes after coming off the bench, giving Peter Gulacsi no chance after the Leipzig goalkeeper had produced a phenomenal save to keep Benjamin Pavard's sublime half-volley out of the top-right corner.
A quick-fire second saw Bayern take the lead in the 34th minute when Muller sent Coman's pass beyond Gulacsi at his near post.
However, it was only in front for 81 seconds, with Kluivert picking out the bottom-left corner after being played onside by Jerome Boateng.
Leipzig regained the lead within three minutes of the restart, Forsberg left in plenty of space to head Angelino's cross past Neuer.
Julian Nagelsmann's side looked to soak up the pressure and see out the win, but it was undone when Coman again picked out Muller for a header that Gulacsi could not keep out.