Leipzig worked tirelessly to thwart the hosts, for whom victory would have opened up a four-point gap after third-placed Borussia Dortmund's 4-3 loss at Bayer Leverkusen on Saturday.
Bayern are bidding to win an eighth straight league title.
They were awarded a second-half penalty which was then reversed when VAR showed Robert Lewandowski had been offside.
Second-placed Leipzig often rode their luck to repeatedly frustrate Bayern, but wasted a series of second-half chances themselves.
"The problem was that both teams could live with a 0-0 draw," admitted Bayern forward Thomas Mueller.
"In the end, neither team wanted to take the absolute risks, which meant we didn't play to our strengths."
Bayern coach Hansi Flick expects this season's title race to go to the wire. "The Bundesliga will be tense until the end," he said.