Augsburg's Florian Niederlechner in action with Bayern Munich's Benjamin Pavard
Niko Kovac's men sit one point adrift of both leaders Borussia Moenchengladbach and second-placed Wolfsburg, who drew 1-1 at RB Leipzig.
Gladbach face a difficult trip to Borussia Dortmund later on Saturday with a chance to move three points clear at the summit.
It was the second league match in succession that reigning champions Bayern had slipped up, having lost 2-1 at home to Hoffenheim before the international break.
Augsburg opened the scoring in stunning fashion, Marco Richter finding the net just 30 seconds after kick-off, the fastest goal of the season so far.
Richter took advantage as Bayern struggled to clear a bouncing ball deep in their own box. The 21-year-old sweetly struck a right-footed volley to grab a rapid lead - just as Augsburg had done in the corresponding fixture last year.
Bayern replied as Robert Lewandowski stretched his scoring streak to eight games in the 14th minute from a Serge Gnabry cross.
After a host of missed chances, former Arsenal winger Gnabry threaded the needle in the 49th minute to put Bayern 2-1 ahead.
But a strong, bullying run in injury time by Sergio Cordova on the wing set up Finnbogason, who finished past Manuel Neuer to earn a point for Augsburg, and put another early dent in Bayern's title defence.
Thomas Mueller missed two chances after coming on as a late substitute after a week of controversy over the fact he has not started a game since September.
Timo Werner scored his first goal in five games for Leipzig in a tight tussle against Wolfsburg at the Red Bull Arena.
It was not enough. Wolfsburg scored a late equaliser when Dutch striker Wout Weghorst finished a William cross to keep the race at the top close.
Wolfsburg are the only unbeaten team in the division.
At the Weserstadion, Werder Bremen opened the scoring at home against Hertha Berlin as Josh Sargent's strike took a useful deflection on its way into the net in the seventh minute.
Dodi Lukebakio hit back in the 70th minute for Hertha to secure a 1-1 draw. He beat three defenders, including Theodor Gebre Selassie twice, in a mazy run from the wing.
Union Berlin claimed just the second win of their debut top-flight season by beating Freiburg 2-0 at a bouncing Stadion an der Alten Foersterei.
Marius Buelter gave them the lead with a long-range strike, his third goal of the season.
A second goal from distance by Marcus Ingvartsen, over substitute goalkeeper Niclas Thiede, making his debut, was enough to secure victory for the home side.
Elsewhere, Rouwen Hennings scored a late winner to seal Fortuna Duesseldorf a 1-0 home win over 10-man Mainz.