The Reds were comfortable winners against the Russian League leaders, managed by former Chelsea manager Guus Hiddink.
The only goal of the game came early in the second period when Downing fired from distance to boost his first-team chances.
Liverpool now has six points from three games ahead of a return trip to Russia in two weeks.
The strength of the opposition — Samuel Eto'o, Christopher Samba and Yuri Zhirkov all started — was reflected in some of the names on Liverpool's team-sheet. There was very little of the mixing and matching of previous Europa League games, with Steven Gerrard, Luis Suarez, Glen Johnson, Daniel Agger and Martin Skrtel all taking to the field.
These players are likely to be asked to play two games in three days, with Liverpool heading to Goodison on Sunday knowing victory is crucial to its Premier League ambitions.
They will go into the derby in confident mood after successive victories.
Suarez was busy from the start. Within 10 minutes he had nutmegged two defenders before being crowded, swivelled past another inside the box before over-hitting his cross, and taken up a fantastic position to collect Oussama Assaidi's cross only to direct his shot too close to Vladimir Gabulov.
With little to do defensively, Johnson made the goalkeeper work with a shot from just inside the area in the 21st minute, while Suarez presented Jonjo Shelvey with a straightforward chance to open the scoring. Running towards the edge of the box, the England youngster elected to shoot first time and saw his blast loop into the Anfield Road end.
Liverpool continued to probe, willed by all but a few hundred of the 39,358 crowd. Assaidi showed tenacity to regain possession in the final third, but his subsequent shot was rushed and easy for Gabulov to scoop.
What of Eto'o? The Cameroonian had scored 42 goals in 92 European games prior to this evening but he was handled with skill by Liverpool's defenders, who have rediscovered an assurance absent at the start of the season. Notable among the back four was 19-year-old Andre Wisdom, whose poise and maturity at full-back could make him hard to displace when Martin Kelly and Jose Enrique return to full fitness.
With Wisdom on the right, Johnson again operated on the left and was afforded time and space to surge forward. A one-two with Suarez produced a clear opening but the ball escaped his control at the vital moment.
The half ended in novel fashion. An Anzhi shot, its first, courtesy of Fedor Smolov. It was ambitious — 35-yards — and it was wide. The Russians had come for a draw.
Half-time brought a change for Liverpool, with Raheem Sterling replacing Johnson, and Downing dropping to full-back.
Agger had tried his luck from distance just before the break — straight after it, Skrtel had a go. The goalkeeper fumbled but recovered in time. Seconds later Shelvey centred as Gerrard sprinted towards the near post, Suarez the far. It was the skipper who connected, stretching to execute a header which landed narrowly wide.
In the 53rd minute the goal finally came when Downing cut in from his flank and smacked right-footed into the bottom corner from 25 yards.
It was the wide man's second goal — he also has three assists — in Europe so far this term. "Stewart Downing, he scores when he wants," chanted the Kop.
Sloppy passing in defence in the 70th minute allowed Anzhi a chance to test Brad Jones at last, Eto'o collecting the ball in a central position. The 31-year-old's powerful effort was too close to Liverpool's stand-in goalkeeper.
Joe Allen replaced Shelvey for the final 10 minutes, a period in which Anzhi increasingly asked questions. Agger cleared near his line as Liverpool thwarted the Russian surge, while the final effort of the night came courtesy of Gerrard, whose Kop-end whack was close to making it 2-0.
Liverpoolfc.com journalists' man of the match: Andre Wisdom.
Photo via LiverpoolFC.com