Liverpool posed the greater threat but were either wasteful in front of goal or kept out by Bayern’s well-marshalled defence.
Sadio Mane missed a handful of first-half chances, most notably dragging a shot on the turn wide from inside the penalty area when unmarked.
Joel Matip also failed to convert Roberto Firmino’s cross from six yards.
Matip almost handed Bayern an away goal early on when his misdirected clearance rebounded to safety off Liverpool goalkeeper Alisson.
The German champions were excellent defensively in the second half, with Mane’s 85th-minute header at the near post the closest Liverpool came to scoring.
What a team effort at Anfield in my ? European match for @FCBayern ?? We’ll be ready for the second leg at our home ??⚽? #fr7? #miasanmia pic.twitter.com/kvPqBw17kE— Franck Ribéry (@FranckRibery) February 19, 2019
The second leg at the Allianz Arena will take place on 13 March.
In Champions League history, 31 sides have drawn the first leg of a knockout match at home 0-0 but only 10 have progressed.
Bayern have lost only two of their past 26 home Champions League games, but Liverpool know if they can score it will give them a huge advantage with away goals counting as double in the event of a draw.
If Liverpool are knocked out, they will rue their misfiring performance at Anfield.
Other than Mane’s late header, which was saved by Manuel Neuer, their only other shot on target came from Mohamed Salah in the first half when he failed to get enough on a brilliant ball over the top by Jordan Henderson to test the Bayern goalkeeper.
There were moments when Liverpool’s trademark one-touch attacking play was evident, but they also failed to make the most of promising opportunities.
Credit must also go to Bayern. They curbed their own attacking instincts to produce a resolute defensive performance, with full-backs Joshua Kimmich and David Alaba and centre-backs Mats Hummels and Niklas Sule impressive.