The visitors enjoyed the first real chance of the match after 22 minutes when Adam Morgan sent Jordan Henderson through in the six-yard box, but the former Sunderland man opted to pull the ball back instead of going for goal and the danger was cleared.
Anzhi's first chance came in the 34th minute when Yuri Zhirkov dispossessed Andre Wisdom before crossing to Samuel Eto'o, but the Cameroonian's strike from point blank range was well saved by Brad Jones.
The hosts took the lead right on the stroke of half-time thanks to a fine individual effort from Lacinda Traore. The Ivorian flicked the ball over Sebastian Coates's head before lofting it over Jones and into the net.
Liverpool looked to respond six minutes after the restart when Joe Cole and Morgan linked well, but the latter snatched at his shot and only found the side netting.
Traore should have doubled his personal tally with 15 minutes left having been found by Rasim Tagirbekov's cross, but his header was straight at Jones.
Suso forced a save from the Anzhi keeper just a minute later before Traore went up the other end and shot just wide of the post as Anzhi held out for the win.