The Germany international had last netted in the German top flight in December 2016, going nine games without a goal, but he ended his drought with a fine finish after the break to seal the win for Carlo Ancelotti's men.
Bayern had reason for optimism heading into the encounter, having won six of their eight Bundesliga games in 2017 and drawing the remaining two, but they also knew Dieter Hecking's men would pose a stern test.
Gladbach had lost just four of their 11 Bundesliga games against Bayern since the start of the 2011-2012 campaign, beating the Bavarian side four times and recording three draws.
There would not be another positive result for Gladbach in this one, though, as Muller eventually made the difference on his 250th league appearance after Robert Lewandowski had wasted three fine chances, with Arjen Robben also unfortunate to hit the woodwork.
This weekend's results see Bayern open up a 13-point lead over RB Leipzig after the second-placed team's defeat to Werder Bremen, with nine games remaining.
Gladbach made an encouraging start to the game and Bayern survived an early scare when Javi Martinez only just managed to prevent Tony Jantschke from firing home at close range, the Spain international making a strong block in the fourth minute.
The visitors slowly grow into the game from there on and Lewandowski should perhaps have opened the scoring after being sent clean through on goal by Franck Ribery, only for the Poland international to aim a tame shot straight at goalkeeper Yann Sommer.
Bayern pinned back Gladbach inside their own half as they went in search of the opener and Robben was unfortunate not to hand his side the lead when he hit the upright with a fine shot from just outside the area, Sommer then pulling off a superb save to deny Lewandowski from the rebound.
Lewandowski threatened a third time early in the second half after a fine low cross from Muller, but the former Dortmund star failed to properly get his foot to it as he aimed yards wide.
Raffael then tried his luck from 20 yards after a rare Gladbach counterattack, the forward blasting a powerful shot just wide of the target.
Bayern continued to dominate proceedings, though, and Muller eventually broke the deadlock in the 63rd minute, beating Sommer from close range after a brilliant ball over the top from Thiago Alcantara.
Hecking brought on Josip Drmic and Thorgan Hazard in an attempt to force an equaliser, but the Bayern defence did well to hold onto their slender lead.
Key Opta stats:
- Bayern fielded their oldest average starting line-up (30 years, 10 days) in the Bundesliga since March 2006 (3-1 win at Duisburg, average age 30 years and 224 days).
- Thomas Muller made his 250th Bundesliga appearance for Bayern and scored his 93rd goal. Only four players have scored more Bundesliga goals for Bayern than Muller.
- Muller has lost none of the 73 Bundesliga games in which he scored (69 wins, four draws) – the longest such streak of a Bundesliga player ever.
- Thiago has had a hand in a personal best 13 goals this season (all comps, six goals and seven assists) after setting up Muller's goal.
- For the first time in 14 years Bayern finished a Bundesliga season without conceding a goal against Borussia Monchengladbach. The last time they had achieved this was in 2002-2003 under Ottmar Hitzfeld (0-0 A, 3-0 H).