The Russian side struck the evening’s only goal in first-half stoppage time when Lacina Traore skilfully spun Sebastian Coates and lifted the ball over Brad Jones.
Liverpool pressed in search of an equalizer after the break but it never came, while Anzhi was guilty of spurning some good openings.
The prematch team news confirmed a first ever senior start for Conor Coady in midfield. The 19-year-old formed part of a youthful Liverpool side which consisted of four teenagers, with Jon Flanagan, Adam Morgan and Andre Wisdom also in Brendan Rodgers‘ XI.
They were joined by more experienced colleagues Joe Cole, Jamie Carragher and Stewart Downing as the Reds looked to take a giant step towards qualification from Group A.
A handful of Liverpool fans made the 1,600-mile journey from Merseyside to Moscow, while Anzhi made a marginally smaller trek across Russia to the capital for the game.
It was the host who created the first effort of the evening. Rasim Tagirbekov‘s left-footer across goal from the edge of the box failed to trouble Jones, however, with five minutes played.
The goalkeeper had to be on his toes moments later to prevent Traore latching onto a through-ball as Anzhi looked to set the early tempo.
But the hosts were afforded a real let off in the 23rd minute when Morgan slipped Jordan Henderson through on goal. He elected to try and pull the ball back to allow the young striker a tap in, but Joao Carlos was able to intercept and a promising opening went begging.
Anzhi retaliated with another spell of pressure and, after Traore had miskicked when dangerously positioned, Samuel Eto’o stung the fingertips of Jones with a well-struck effort from 12 yards after Yuri Zhirkov had stolen possession from Wisdom.
Chances were proving tough to muster for both sides, but it took a moment of real skill to break the deadlock in first-half stoppage time.
Traore’s clever flick-and-spin beyond the attentions of Coates sent him bearing down on goal. With Jones rushing out to meet him, the striker lofted the ball over the advancing No. 1 to alter Rodgers’ half-time team-talk.
With the Reds eager to conjure up a speedy riposte, Morgan struck into the side-netting after a neat pass from Cole released him into space. Meanwhile, at the other end Jones was required to pluck Eto’o's curler out of the air while Traore was millimeters away from grabbing a second with a 20-yard blast.
With an hour played, Rodgers opted to inject a Spanish influence into his side with Iberian duo Dani Pacheco and Suso taking to the field to replace debutant Coady and Morgan.
Liverpool began to take control of possession, though Jones was needed to thwart Traore’s close-range header as the game edged towards a conclusion.
Suso’s introduction had provided the Reds with a different attacking impetus, and the 18-year-old was involved in their best opportunity of the evening so far after 77 minutes. His shimmy on the edge of the box created space for a shot, which was spilled by Vladimir Gabulov — but the Anzhi custodian sprung to his feet swiftly to repel Cole’s follow-up.
With the visitors pushing forward in search of an equalizer, a counter-attack from Guus Hiddink‘s men resulted in Traore racing through on goal, only to steer wide.
The Reds continued to toil throughout the final stages in the hope of clinching a point, but it wasn’t to be as Anzhi held out.
Liverpoolfc.com journalists’ man of the match: Suso.