Guus Hiddink's side started rather cautiously, despite having home advantage against a very inexperienced team, comprised mostly of fringe and youth-team players.
However, the England international chose to cut the ball back instead of shooting at goal, but only succeeded in finding Christopher Samba, who hacked clear.
The home side have struggled to penetrate the Liverpool defence, which has looked well organised and composed on the ball. Brendan Rodgers's men have defended in numbers and allowed their opponents very little space in the final third.
The former Chelsea man then squared the ball to Samuel Eto'o inside the box, but the Cameroonian's effort was pushed over by the Australian goalkeeper.
However, a piece of individual brilliance from Lacina Traore gave Anzhi the lead with the last action of the half.