Bayern chairman Karl-Heinz Rummenigge ruled out selling Robben in the summer, even if the club fail to qualify for the Champions League next season.
And the Holland winger responded with the only goal of the game after 77 minutes as the home side took advantage of Hannover's 4-1 thrashing at Borussia Dortmund to climb above them in the table.
There were few chances in the first half, but all of the clear ones fell to Bayern.
Robben flashed a 17th-minute free-kick over the bar and two minutes later brought a save from Gladbach goalkeeper Logan Bailly.
However, Bailly could only parry the ball back into the centre of the box, where Mario Gomez was only inches away from connecting with the ball to open the scoring.
A minute later Robben headed over from Franck Ribery's cross as the home fans began to think it might not be their day.
Bayern continued to make the running in the second half and Ribery flashed a shot inches wide of the post.
The same player then fired a free-kick narrowly over before the visitors enjoyed a brief spell of pressure.
Marco Reus looked to have almost run the ball out of play in the 69th minute before firing in a shot from an acute angle which clipped the bar, and five minutes later Mohammadou Idrissou's shot was blocked by goalkeeper Thomas Kraft.
However, Bayern finally broke the deadlock on 77 minutes as Robben broke from halfway before passing to Ribery.
Ribery reached the byline before cutting the ball back for Robben to control and fire left-footed into the bottom corner.