Wearing replicas of the first Bayern shirts 120 years ago, to mark the club's birthday, they laboured to add another in the second half with Augsburg keeper Andreas Luthe stopping a point-blank Philippe Coutinho shot and Serge Gnabry scuffing a handful of golden chances.
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Augsburg wasted their best opportunities late in the game when Florian Niederlechner's shot was blocked by Manuel Neuer and Alfred Finnbogason headed wide in the 81st.
Bayern's Leon Goretzka then slotted in their second in stoppage time to kill off the game.
Bayern move up to 55 points, four ahead of Borussia Dortmund, who are second after Saturday's 2-1 win at Borussia Moenchengladbach. RB Leipzig are third following a goalless draw at VfL Wolfsburg.