Nagelsmann became a source of anger among Leipzig fans after he departed for Bayern and then also took Marcel Sabitzer with him, and any such ill-feeling will not have eased after this victory.
Robert Lewandowski put Bayern in front early on with a well-taken penalty and then a quick-fire double courtesy of Jamal Musiala and Leroy Sane early in the second half put the game beyond the hosts.
Jesse Marsch's men quickly pulled one back in spectacular fashion through Konrad Laimer, but it was not enough to inspire a comeback as Leipzig's miserable record against Bayern continued, Eric Maxim Choupo-Moting adding a stoppage-time fourth for good measure.
Leipzig had already seen a penalty shout of its own controversially ignored when Kevin Kampl was penalised for handball after a video assistant referee (VAR) check in the 12th minute, with Lewandowski sweeping home the resulting spot-kick.
The hosts grew as a threat as the first half progressed and Bayern received a blow just before half-time when Serge Gnabry was forced off.
But his replacement quickly made an impact just after the break, as Musiala volleyed home Alphonso Davies's cross.
The misery of Andre Silva's disallowed goal shortly after was compounded by Sane burying a Musiala cross to essentially end Leipzig's hopes of a turnaround.
It certainly showed spirit as Laimer got one back with a sumptuous long-range strike and Silva almost reached a Christopher Nkunku pass for a tap-in.
But it was Bayern that got the only late goal, Choupo-Moting showing good composure to slam past Peter Gulacsi after Kimmich sliced open the Leipzig defence.