Robert Lewandowski calmly slotted home from the spot after 13 minutes at Augsburg Arena and missed a glut of chances in a first half dominated by the champions.
It was a completely different outlook for Bayern after the break, though, and Hansi Flick's side were left breathing a sigh of relief when Alfred Finnbogason struck the post with a penalty of his own 15 minutes from time.
The end result means Bayern remain four points clear at the top but Flick will want to see fewer of the Jekyll-and-Hyde performances that have been on display in recent weeks from his side.
Rani Khedira's clumsy kick on Lucas Hernandez allowed Lewandowski to find the bottom-right corner with a cool 13th-minute penalty that looked like setting the tone for a dominant win.
Serge Gnabry was denied by the busy Rafal Gikiewicz and Thomas Muller put the rebound narrowly wide as Bayern peppered the hosts' goal.
Gnabry was kept out again by Gikiewicz before teeing up Lewandowski, who was guilty of wastefulness a couple of times, with the Pole only hitting the outside of the post with the goal gaping.
A second half completely at odds with the first saw Bayern struggle to create openings and Daniel Caligiuri's long-range effort would have briefly had away hearts in mouths before rising comfortably over.
There was an even greater left-off shortly after when substitute Finnbogason fluffed his lines from the spot after Benjamin Pavard had been pinged for handball from Iago's flick in the area.
Muller had the ball in the net but Jamal Musiala clearly ran the ball out in the act of setting him up, before Florian Niederlechner skewed wide and Felix Uduokhai headed over in further late scares that a lacklustre Bayern ultimately survived.