The Russians are among the biggest wage-payers in football, with Cameroon international Eto'o reputedly earning £250,000 a week playing for the Dagestan-based side.
But having fielded a side close to his strongest, Rodgers watched them comfortably secure a win that will give them a confidence boost for Sunday's derby clash with Everton.
They were always in control, creating a steady stream of chances throughout. The first of them fell to Jonjo Shelvey but he wasted an opportunity to add to his three goals in the competition.
Luis Suarez got away from defender Joao Carlos to square for the midfielder, only for Shelvey to scoop his right-footed effort over the bar from the edge of the area.
Suarez was involved again as Liverpool opened up the Russians once more, spotting Glen Johnson's rare foray into the box and delivering a clever flicked pass into his path that the left-back failed to make contact with.
Anzhi, with former Blackburn defender Chris Samba at the heart of their defence, came under further pressure as half-time approached.
An off-target Nuri Sahin header and a shot wide from Agger suggested a growing urgency in Liverpool's play.
Agger's defensive partner Martin Sktrel was next to test Gabulov with a shot from distance as the hosts continued to dominate possession.
And it was a surprise when Steven Gerrard passed up a chance to score the opener.
Half-time substitute Raheem Sterling beat Samba in the air to guide the ball to Shelvey on the left, and the latter's first-time cross found Gerrard unmarked but the captain sent his header wide of the near post.
That miss was quickly forgotten however as Downing, having been moved to left-back following the introduction of Sterling, struck in the 53rd minute.
He strode onto a Shelvey pass and used his "wrong" foot to drill the ball across goal and into the far corner of the net.
Suarez went close to adding a second moments later with a fierce drive that flew just past the upright before Agger saw his effort ruled out after he nodded the ball off the keeper's outstretched hand.
Anzhi, who were unbeaten in 13 games coming into this match, had only an Eto'o effort to cheer as the Reds closed out the match in convincing style.
Full Time The referee signals the end of the game.
Free kick awarded for an unfair challenge on Mehdi Carcela-Gonzalez by Nuri Sahin. Mehdi Carcela-Gonzalez takes the direct free kick.
Free kick awarded for a foul by Yury Zhirkov on Luis Suarez. Luis Suarez restarts play with the free kick.
Mehdi Carcela-Gonzalez produces a left-footed shot from outside the penalty area and misses left.
Short corner taken by Luis Suarez.
Mehdi Carcela-Gonzalez takes a shot. Comfortable save by Brad Jones.
Shot by Steven Gerrard from outside the box goes over the net.
Yury Zhirkov concedes a free kick for a foul on Raheem Sterling. Martin Skrtel takes the free kick.
Andre Wisdom produces a cross.
Oussama Assaidi delivers the ball.
Effort from close range by Lacina Traore goes wide left of the target.
Effort from outside the area by Raheem Sterling goes wide right of the goal.
The referee penalises Mehdi Carcela-Gonzalez for handball. Steven Gerrard takes the direct free kick.
Oleg Shatov takes a shot. Andre Wisdom gets a block in.
Yury Zhirkov fouled by Raheem Sterling, the ref awards a free kick. Oleg Shatov restarts play with the free kick.
Booking Daniel Agger is given a yellow card.
Inswinging corner taken by Nuri Sahin, Daniel Agger takes a shot. Vladimir Gabulov makes a save.
Substitution Joe Allen comes on in place of Jonjo Shelvey.
Corner taken right-footed by Oleg Shatov from the left by-line to the near post, clearance made by Nuri Sahin. Corner taken by Oleg Shatov from the left by-line played to the near post, clearance by Andre Wisdom. Corner taken by Oleg Shatov, Brad Jones makes a comfortable save.
Substitution (Anzhi Makhachkala) makes a substitution, with Shamil Lakhiyalov coming on for Mbark Boussoufa.
Mbark Boussoufa takes a shot. Blocked by Andre Wisdom.
Vladimir Gabulov takes the direct free kick.
Booking Luis Suarez is cautioned.
Luis Suarez challenges Mehdi Carcela-Gonzalez unfairly and gives away a free kick.
Mbark Boussoufa gives away a free kick for an unfair challenge on Luis Suarez. Steven Gerrard takes the direct free kick.
Free kick awarded for a foul by Arseni Logashov on Oussama Assaidi. Direct free kick taken by Stewart Downing.
Samuel Eto'o takes a shot. Save by Brad Jones.
Free kick awarded for an unfair challenge on Oleg Shatov by Nuri Sahin. Free kick crossed by Mbark Boussoufa, clearance by Steven Gerrard.
Mbark Boussoufa gives away a free kick for an unfair challenge on Stewart Downing. Steven Gerrard restarts play with the free kick.
The ball is swung over by Arseni Logashov, Daniel Agger makes a clearance.
Samuel Eto'o takes a shot. Daniel Agger gets a block in.
Substitution Fedor Smolov leaves the field to be replaced by Lacina Traore.
The ball is crossed by Stewart Downing, Mehdi Carcela-Gonzalez makes a clearance.
Steven Gerrard takes a inswinging corner from the left by-line to the near post, Yury Zhirkov makes a clearance.
Indirect free kick taken by Vladimir Gabulov.
Substitution (Anzhi Makhachkala) makes a substitution, with Arseni Logashov coming on for Kamil Agalarov.
Raheem Sterling is ruled offside.
Free kick awarded for a foul by Georgi Gabulov on Jonjo Shelvey. Stewart Downing takes the direct free kick.
Jonjo Shelvey takes a shot. Vladimir Gabulov makes a comfortable save.
Martin Skrtel concedes a free kick for a foul on Fedor Smolov. Mbark Boussoufa takes the free kick.
Luis Suarez has an effort at goal from outside the penalty box missing to the wide left of the goal.
Foul by Fedor Smolov on Raheem Sterling, free kick awarded. Free kick taken by Jonjo Shelvey.
Samuel Eto'o takes a shot. Save made by Brad Jones.
Assist on the goal came from Jonjo Shelvey.
52:45 Goal scored
Goal - Stewart Downing - Liverpool 1 - 0 Anzhi Mak Stewart Downing fires in a goal from long range to the top right corner of the goal. Liverpool 1-0 Anzhi Makhachkala.
Yury Zhirkov challenges Raheem Sterling unfairly and gives away a free kick. Steven Gerrard takes the direct free kick.
Free kick awarded for an unfair challenge on Jonjo Shelvey by Fedor Smolov. Brad Jones takes the direct free kick.
Headed effort from deep inside the penalty area by Steven Gerrard misses to the left of the goal.
Stewart Downing delivers the ball. Martin Skrtel takes a shot. Save by Vladimir Gabulov.
Andre Wisdom concedes a free kick for a foul on Fedor Smolov. Vladimir Gabulov takes the free kick.
Unfair challenge on Fedor Smolov by Andre Wisdom results in a free kick. Free kick taken by Mbark Boussoufa.
The referee starts the second half.
Substitution Raheem Sterling on for Glen Johnson.
45:00 +1:02 Half time
Half Time It is the end of the first-half.
Long range shot by Fedor Smolov goes wide right of the net.
The free kick is delivered right-footed by Steven Gerrard from left wing, Yury Zhirkov manages to make a clearance.
Booking The referee books Kamil Agalarov for unsporting behaviour.
Free kick awarded for a foul by Kamil Agalarov on Oussama Assaidi.
Long range effort from Daniel Agger misses to the left of the goal.
Steven Gerrard takes the outswinging corner, Header from deep inside the area by Nuri Sahin goes harmlessly over the crossbar.
Pinto Joao Carlos gives away a free kick for an unfair challenge on Stewart Downing. Free kick crossed left-footed by Nuri Sahin, save made by Vladimir Gabulov. Free kick awarded for an unfair challenge on Oleg Shatov by Steven Gerrard. The free kick is delivered right-footed by Mbark Boussoufa from right wing, Steven Gerrard makes a clearance.
Oussama Assaidi takes a shot. Save made by Vladimir Gabulov.
The referee gives a free kick against Mehdi Carcela-Gonzalez for handball. Direct free kick taken by Daniel Agger.
Effort on goal by Jonjo Shelvey from just inside the area clears the bar.
Foul by Fedor Smolov on Andre Wisdom, free kick awarded. Andre Wisdom restarts play with the free kick.
Steven Gerrard shoots direct from the free kick.
Booking The referee shows Christopher Samba a yellow card for unsporting behaviour.
Christopher Samba gives away a free kick for an unfair challenge on Luis Suarez.
Glen Johnson takes a shot. Save by Vladimir Gabulov.
Oussama Assaidi sends in a cross, clearance by Christopher Samba.
Luis Suarez takes a shot. Comfortable save by Vladimir Gabulov.
The ball is swung over by Luis Suarez.
The assistant referee signals for offside against Jonjo Shelvey. Pinto Joao Carlos takes the indirect free kick.
Outswinging corner taken by Nuri Sahin.
The ball is crossed by Andre Wisdom, clearance by Christopher Samba.
The match gets underway.
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