The Bavarians were unchanged from the last time they were involved which was against Hertha Berlin, but Dani Olmo picked out Leipzig's starting eleven.
Bayern seized control 12 minutes in when Thomas Muller attacked a loose ball in the area following a spill by Peter Gulacsi to thump in from close-range.
Olmo almost led the visitors back on equal terms minutes thereafter through a one-on-one from a ball into space by Christopher Nkunku, but goalkeeper Manuel Neuer smothered.
Though, the equaliser arrived for the visitors three minutes to the half-hour mark when Andre Silva latched onto a teasing attempt by Konrad Laimer and nudged in from close-range.
Bayern could've reclaimed the lead through a cross-cum shot by Kingsley Coman in the last few minutes of the half, but Gulacsi palmed onto the woodwork.
The host did regain their advantage inside 44 minutes when Robert Lewandowski met a cross from Coman to head beyond the reach of Gulacsi.
As Dominik Szoboszlai replaced Kevin Kampl for Leipzig, the visitors nabbed the equaliser eight minutes in the second half after Nkunku ran onto Laimer's pass to apply a composed finish.
Julian Nagelsmann's men restored their cushion after a few minutes through Josko Gvardiol who helped a teasing attempt from Serge Gnabry past Gulacsi and into his own net.
Lewandowski could've had a brace and put Bayern further ahead with a header late on from a ball by Gnabry, but he went wide of target.
The Bavarians almost surrendered the lead again when Emil Forsberg got two chances within seconds, but Neuer collected to give the host victory and three points.