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The pressure has been building on Kovac, and his side shipped five more goals on a day that saw Thomas Muller make his 500th appearance for the club.
Filip Kostic and Djibril Sow scored their first goals of the season to punish Bayern after Boateng was sent off inside nine minutes.
The German champion threatened a comeback through Lewandowski's 14th league goal of the season, but David Abraham, Martin Hinteregger and Goncalo Paciencia struck in the second half to make it one league win in four matches for the big-name visitors, who have slipped to fourth in the table.
Boateng was dismissed for a trip on Paciencia after a VAR (video assistant referee) check from referee Markus Schmidt, who initially showed a yellow card and awarded a penalty before amending it to a red and a free-kick.
The hosts made the man advantage count 16 minutes later when Sow's shot struck David Alaba and landed in the path of Kostic for a simple finish from close range.
A bad day for Bayern became even worse just after the half-hour mark as a brilliant passing move culminated in Sow calmly converting Kostic's deflected cross at the back post.
Joshua Kimmich had a goal ruled out for offside but Bayern had one back before the interval, Lewandowski spinning away from three men and converting past Frederik Ronnow.
Any hope of a comeback was killed off by Abraham four minutes into the second half, however, with the Frankfurt skipper stabbing home Danny da Costa's inch-perfect cross.
Abraham's central-defensive team-mate Hinteregger headed in a fourth from a corner after 61 minutes and, after Alphonso Davies clipped the post, Paciencia rounded off the scoring late on following some good play from substitute Andre Silva.
Having already been publicly criticised by club CEO Karl-Heinz Rummenigge last month, this third loss of the campaign in all competitions — against his former side — could prove the final straw for Kovac.
The big matches maintain the pressure, too, with Bayern hosting Olympiacos in the UEFA Champions League on Thursday (AEDT), before meeting title rival Borussia Dortmund in a week's time in a huge Bundesliga showdown.
