Bayern, which has lost only once in the league all season, took a big step toward an unprecedented fifth straight title after Leipzig's surprise defeat at Werder Bremen on Saturday.
Arjen Robben tested Gladbach goalkeeper Yann Sommer inside the first minute with a low drive.
Thiago Alcantara was the architect for Muller's victor, flicking his pass over the Gladbach defence for the Germany forward to fire home his 93rd Bundesliga goal on 63 minutes.
It was the Germany forward's 93rd Bundesliga goal for Bayern in his 250th appearance in Germany's top flight, but only his second in the Bundesliga this season.
Mid-table Monchengladbach were penned back for long periods and often seemed non-existent as an attacking force.
Wes Hoolahan ruled out of Ireland's World Cup qualifier against Wales
He thinks that he might be able to make it - that would be great if he can. He said: "Seamus has picked up a knock - it's precautionary".
Schalke 04 climbed two places to ninth as masked Bosnia defender Sead Kolasinac produced a superb individual goal after 50 minutes in their 1-0 win at Mainz.
Carlo Ancelotti discusses how he underestimated Monchengladbach, and the issues they created for Munich to win. Robben unleashed a trademark curling effort that cannoned off the angle of post and crossbar and Lewandowski's diving header on the follow-up was brilliantly saved by the home keeper.
Bayern were not at their best, with top scorer Robert Lewandowski heading over from Muller's cross early on, and skewing a shot wide when well placed after half-time.
Bayern's profligacy was nearly punished when Raffael fired just wide, but Muller finally gave the visitors the lead a minute later.
Manuel Neuer came to Bayern's rescue after the hour mark, as he rushed off his line to deny Andre Hahn a clear sight of goal.