Held to a 1-1 home draw by Augsburg in midweek, the return of Robert Lewandowski, James Rodriguez and a raft of regulars failed to prevent Bayern from dropping points for a second successive outing.
They were disjointed in defence and uncharacteristically wasteful in attack as Vedad Ibisevic's 23rd-minute penalty and a shot from Ondrej Duda on the stroke of half-time gave the home side their first victory against the Bavarian giants in 15 league meetings.
It added up to a forgettable homecoming for former Hertha player Kovac, who will expect a response to this mini slump when Bayern host Ajax in the Champions League on Tuesday.
Recalled as one of seven changes, Jerome Boateng's first major act was to nod a James corner agonisingly wide of the top corner.
His next, however, was a panicked challenge on Salomon Kalou that allowed Ibisevic to convert the opener, leaving Bayern rattled on the road.
Arjen Robben inexplicably turned David Alaba's low cross over the bar at point-blank range and the miss was punished a minute before the break, Duda emphatically meeting Valentino Lazaro's clever cut-back.
Boateng went close to reducing the deficit with another fine header beyond the hour but Thomas Kraft, deputising for the injured Rune Jarstein, reacted well to deny the Germany international.
Kovac threw on Thomas Muller, Serge Gnabry and Sandro Wagner in an ultimately fruitless search for a way back into his most alarming match in charge.
What it means: Rotate with caution
Bayern had won four straight and scored 11 in the Bundesliga prior to Kovac's decision to field a vastly altered side against Augsburg.
Held to a point on that occasion, they were defensively disorganised at Olympiastadion despite the return of their leading names, with the regular rotation at the back between Boateng, Mats Hummels and Niklas Sule doing no favours for any of the trio and potentially requiring a rethink, at least in the early part of the campaign.