Goals were predicted on Merseyside in a battle between two five-time champions of this competition, but shots on target were few and far between on a night of little drama.
Joel Matip squandered the best opportunity of the match in the first half and Sadio Mane also came close on a couple of occasions, though the Reds could not find a breakthrough and now have a tough task in the reverse fixture in three weeks' time.
Bayern had failed to score on their previous three visits to Anfield and they lacked a cutting edge throughout the contest.
The visitors' best opportunity of the contest came via Liverpool defender Matip as he directed Gnabry's cross towards his own goal, but it ricocheted off Alisson's face in a real let-off.
Liverpool created more opportunities at the other end but failed to convert any of them, with Mohamed Salah shooting straight at Manuel Neuer when getting on the end of a fine deep pass from Jordan Henderson, before heading wide after ghosting in at the back post.
Mane also sent a shot wide on the spin and was wayward with another, while Matip slotted a glorious opportunity the wrong side of Neuer's post in the best of the game's chances.
Not since April 2017 had Bayern failed to attempt a single shot on target in the first half of a Champions League match, and they did not improve much in that regard in the second 45.
Liverpool continued creating the odd chance, seeing Salah squander a decent opening when Andrew Robertson's pass made it into his path at the back post.
Gnabry then had Alisson scrambling with his shot from 20 yards as Bayern focused on counter-attacks, while at the other end of the field the English hosts were growing increasingly frustrated.
James Rodriguez and Robertson both flashed shots well wide, but the closest either side came to a late winner was Liverpool when Mane got in front of his man and nearly picked out the bottom corner with his header, forcing Neuer into a smart save down low.
LIVERPOOL (4-3-3): Alisson; Alexander-Arnold, Matip, Fabinho, Robertson; Wijnaldum, Henderson, Keita (Milner 76'); Mane, Salah, Firmino (Origi 76')
BAYERN MUNICH (4-2-3-1): Neuer; Kimmich, Sule, Hummels, Alaba; Thiago, Martinez; Gnabry (Rafinha 91'), Rodriguez (Sanches 88'), Coman (Ribery 81'); Lewandowski