The Bundesliga champion also lost Niklas Sule to an injury that cause some concern for coach Niko Kovac as he heads into a UEFA Champions League week.
Richter scored in the opening minute for host Augsburg as Bayern struggled to bounce back from the loss suffered to Hoffenheim before the international break.
But after the visitors lost a distressed Sule to injury, Serge Gnabry set up Robert Lewandowski to net for a record-equalling eighth straight game at the start of a Bundesliga campaign.
Former Arsenal youngster Gnabry then scored himself in the second half.
That would not prove to be the winner, though, Alfred Finnbogason tapping in one minute into added time to stun Niko Kovac's side.
Had Bayern won, it would have returned to the top of the Bundesliga after Wolfsburg lost at RB Leipzig. Instead, Bayern was left a point off the pace, with Borussia Monchengladbach in with a chance to extend its lead.