A win would have made Bayern champions for the 26th time but that was denied them by Andre Hahn's equaliser, driven low into the net 18 minutes from time.
Thomas Muller headed Bayern in front with a 20th league goal of the season.
Borussia Dortmund beat Wolfsburg 5-1 to move five points behind Bayern with two games left.
Bayern face a local derby at Ingolstadt next weekend before playing Hannover at home in their final game.
Bayern coach Pep Guardiola made eight changes to the side that lost 1-0 in the first leg of their Champions League semi-final to Atletico Madrid with the home leg to come next Tuesday.
One of those was the return of Muller. Bayern had won 51 consecutive league games in which the Germany international has scored but Hahn's fifth goal of the season ended that run.
Bayern face Dortmund in the German Cup final on 21 May as they look to complete a treble under Guardiola before his summer move to Manchester City.
Gladbach ended a run of four successive away defeats to enhance their chances of qualifying for the Champions League as they moved above Hertha Berlin, who lost 2-1 at third-placed Bayer Leverkusen, into fourth place on goal difference.
Bayern Munich coach Pep Guardiola: "Of course we had wanted to win today, for our fans and for ourselves.
"We need just a point now, a draw, or a win to secure the title.
"But now there's time to concentrate on Atletico Madrid. It will be a different game on Tuesday. We must play better."
Bayern chairman Karl-Heinz Rummenigge: "We would have liked to be champions today, I won't lie. But we'll try to seal the deal against Ingolstadt next weekend.
"The spirit was there, but you can't forget that we had a very difficult game three days ago.
"Now we switch all our focus towards Tuesday. I hope that we can pull it off.
"We will throw everything at the game and put up a fight, with 70,000 fans behind us. I can promise that."
Bayern Munich must wait to secure a record fourth successive Bundesliga title after being held to a 1-1 draw by Borussia Monchengladbach.