Mueller sewed up his brace and ensured the 6-0 victory after making use of Jerome Boateng's cross into the box with ten minutes to play.
With the result, Bayern secured the fifth straight win in the league to tighten their top spot in the standings with 65 points. Meanwhile, Augsburg slipped from the 14th to the 16th place, which is a relegation play off spot.
Second placed Leipzig defend their position in the standings as Naby Keita's brace helped the "Bulls" to cruise 4-0 past bottom dwelling Darmstadt.
Leipzig caught a bright start on home soil as Naby Keita broke the deadlock with 12 minutes into the encounter when he volleyed home a clearance from inside the box.
Despite the setback Darmstadt tried to restore parity but neither Sidney Sam nor Fabian Holland were able to make the most of their promising opportunities.
Ralph Hasenhuettl's men punished Darmstadt's wastefulness as Emil Forsberg drilled the ball from the edge of the box into the bottom right to double the lead in the 67th minute.
The hosts put the result beyond doubt in the 79th minute when Willi Orban headed home a corner before Keita sealed his brace and the 4-0 victory moments later.
With the 16th win of the season, Leipzig consolidated their second place with 52 points whereas last-placed Darmstadt suffered the 19th loss of the season.
Elsewhere, third-placed Dortmund were held to a 1-1 draw by rivals Schalke at the 171st "Revierderby".
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