Goals from Thiago and Xabi Alonso put the home side firmly in control before Leipzig’s Emil Fosberg saw red for a nasty foul on Philipp Lahm.
Robert Lewandowski made it 3-0 from the penalty spot after Leipzig goalkeeper Peter Gulasci had fouled Douglas Costa.
Lewandowski was denied twice as Leipzig kept Bayern at bay after the break.
It is just a second league loss of the season for Leipzig, who are in their debut Bundesliga campaign.
Formed in 2009 when drinks manufacturer Red Bull bought the licence of amateur club SSV Markranstadt and changed their name, Leipzig have achieved four promotions in seven years to reach the German top flight.
Under coach Ralph Hasenhuttl, they won 11 of their first 15 matches, including a victory over last season’s runners-up Borussia Dortmund.