It was a 28th win in 33 league games, with one match left in the season, for sensational Bayern who weeks ago secured their 23rd German league title and 22nd since the introduction of the Bundesliga half a century ago. Augsburg's Alexander Manninger made two sensational first- half saves to deny Mueller and Mario Mandzukic from point-blank range and the battling visitors even hit the bar early in the second half.
But Austrian Manninger was well beaten when Brazil international Dante floated a long cross into the box and Mueller rose high at the far post to head in.
Franck Ribery took on four Augsburg players and threaded the ball to Shaqiri who turned and scored in the 81st before Brazilian Gustavo completed the win with a low drive three minutes from time.
Dortmund were far less convincing, needing two late goals from Marco Reus to escape with a 3-3 draw at VfL Wolfsburg.
At the other end of the table, Hoffenheim's chances of staying up all but vanished with a 4-1 demolition by visiting Hamburg SV.