The hosts were far superior in the first half but were unable to get the ball in the Russian net.
© PA Photos
Anzhi's goalkeeper was called into action again soon after as he beat away Glen Johnson's drive.
The hosts were soon in control and Jonjo Shelvey should have done better midway through the half when he blasted over, despite having plenty of time and space to get a shot away.
Liverpool's best opportunity came with 10 minutes left to play as Johnson lost his footing at the crucial time when in on goal. The defender claimed that he had been tripped by the Anzhi defender but the referee dismissed the appeals.