Despite the absence of key players, including the injured Franck Ribery and Bastian Schweinsteiger, treble winners Bayern picked up where they left off in December to stretch their unbeaten run to 42 consecutive league games.
"I am happy for my team," Bayern coach Pep Guardiola told reporters. His team had suffered a surprise 3-0 defeat away to Salzburg in a friendly game last week.
"Salzburg was a good lesson for us. Tonight we worked hard and did it well," said the Spaniard. "We created many chances. Borussia are an incredibly intelligent side and I am very happy with our performance."
After a nervous opening spell in which third-placed Gladbach's Max Kruse squandered a golden chance, Bayern, who have now won their last nine league games, settled and came close when Mario Goetze's shot scraped the post.
Midfielder Goetze started in a lone striker role after Mario Mandzukic was dropped from the squad for not having trained well, according to sports director Matthias Sammer.
Mandzukic, last season's first choice forward, has lost his regular starting spot under Guardiola, who will also have Poland striker Robert Lewandowski in his squad from next season.
Goetze soon had better luck when Mueller raced down the right wing and cut the ball back for his Germany team mate to drill into the net for the lead after seven minutes.
Gladbach were constantly on the backfoot against a Bayern side who never hit top speed but controlled possession, using defender Philip Lahm as a holding midfielder while Spaniard Thiago orchestrated the game.
However, the hosts did come close twice on the stroke of halftime with Bayern goalkeeper Manuel Neuer palming a Kruse shot on to the post and denying Juan Arango from close range.
Mueller got on the scoresheet eight minutes after the restart, converting a penalty to condemn Gladbach to their first home defeat of the campaign before the hosts hit the woodwork again through Patrick Herrmann.
On Saturday, second-placed Bayer Leverkusen, on 37 points, are in action at lowly Freiburg while Borussia Dortmund, fourth with 32, host mid-table Augsburg. Gladbach are third on 33.
(Reporting by Karolos Grohmann; editing by Ken Ferris)
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