The Bavarians would have secured a seventh successive title with a win, or if they bettered Borussia Dortmund's result against Fortuna Dusseldorf, but the draw and a 3-2 BVB win mean the title race goes down to the final day.
Despite a number of chances, Bayern were unable to break down the organised Leipzig defence to find the winner to secure the title.
Leipzig started the game strongly, with Yussuf Poulsen coming closest in the opening 20 minutes, but the striker's shot was well blocked by Mats Hummels.
After a spell of pressure from the home side, Bayern started making their mark on the game, with Serge Gnabry prevented from taking a shot by Leipzig's defenders.
The former Arsenal player came close again in the visitors' best chance of the half, volleying from the edge of the box but his effort was saved by goalkeeper Peter Gulacsi.
Bayern had the ball in the back of the net following Leon Goretzka's powerful volley into the net but the goal was ruled offside following a VAR review.
David Alaba beat the defender at the near post to cross for Thomas Muller but the striker's headed effort was just off target.
As Bayern increased the pressure on Leipzig, Gnabry had another chance, with only the keeper to beat, the 23-year-old's effort rebounded off the crossbar.
Against the run of play, Leipzig had an opportunity to take the lead, Poulsen played in Marcel Halstenberg whose effort was dragged wide from a tight angle.
Robert Lewandowski came closest to scoring the deciding goal in the final stages of the game, the Polish striker curled his free-kick round the wall but narrowly missed the goal, with Gulacsi unmoved in the Leipzig net.
In one of his final matches for the Bavarian side, Arjen Robben also came close, striking the ball from outside the area but he was unable to get on top of it and his effort went over the bar.