Bayern came into the contest looking to take advantage of Borussia Dortmund’s 2-2 draw against Werder Bremen on Friday evening.
However, those hopes were dealt the earliest of early blows when Manuel Neuer, receiving the ball from kick-off, gave it away in costly fashion as Fellhauer chased down his attempted pass and rolled the ball into the awaiting net inside eight seconds.
Augsburg were then forced to soak up a long period of Bayern pressure before they threatened to add a second from a corner, only for Juma Bah to scoop his close-range opportunity over the bar when it appeared easier to hit the target.
Despite that gilt-edged miss, Augsburg scraped through to the break with a hard-fought lead, but not before Harry Kane struck the woodwork with the final chance of a gripping first period.
Determined to spark a turnaround, Bayern came out of the half-time break with real purpose and soon found a way past Finn Dahmen, who failed to keep out Luis Diaz’s acrobatic strike despite getting two hands to the ball.
The goalkeeper’s blushes were ultimately spared by an offside in the build-up, but any sense of relief was soon quashed by Ismael Saibari's legitimate equaliser just moments later.
Collecting Kane’s precise ball, the Morocco international slotted into the bottom corner to score in back-to-back Bundesliga games.
Rather than take a step back, Manuel Baum’s fearless charges restored the lead through Rodrigo Ribeiro, who was left with a simple close-range finish after Neuer parried Calvin Brackelmann’s powerful drive.
Having been dealt another setback, the Bavarians’ frustration grew when Serge Gnabry and Diaz had potential levellers ruled out in quick succession for offside infringements.
However, Kane eventually helped end Augsburg’s latest resistance, neatly combining with Aleksandar Pavlovic before teeing up Nathaniel Brown for a tap-in.
With the hosts rocking, Gnabry went agonisingly close to snatching the spoils with two efforts that struck the woodwork, including one that may have drifted over the line, but an offside call ultimately fell in Augsburg’s favour, leaving the visitors to settle for a draw that keeps them two points adrift of leaders Dortmund.
Flashscore Man of the Match: Finn Dahmen (Augsburg)