Lennart Karl opened the scoring for the German champions before Maurice Krattenmacher brought Elversberg level. Harry Kane then restored Bayern's lead as the visitors responded strongly after being pegged back.
Bayern head coach Vincent Kompany admitted afterwards that his team took too long to find their rhythm and were not at their usual level in the first half.
"It felt like we had a slightly harder time getting into the game today," Kompany told DAZN.
The Belgian felt Bayern lacked both precision and intensity, allowing Elversberg to grow in confidence.
"We simply weren't there at the beginning. They played well, they were courageous, and we weren't as precise, even on the ball, as we usually are," he said.
Kompany also highlighted Bayern's pressing as an area that needed improvement.
"If your counter-pressing isn't intense enough, then it's a game like this; you don't get any gifts here," he added.
The Bayern boss was pleased with the way his players reacted after Elversberg equalised, but warned that they cannot afford similar slow starts against stronger opponents.
"We were a step too slow mentally at the beginning. That makes things difficult for ourselves," Kompany said.
Despite the problems, Bayern showed their character to regain control and leave with three points.
"In the end, we redeemed ourselves," Kompany added.
Bayern now return to the Allianz Arena on Friday, when Union Berlin provide their next Bundesliga test.
