The Reds dominated large spells of the first half without managing to find a route to goal, but a half-time switch brought quick rewards for Brendan Rodgers' men.
Stewart Downing, moved to left-back from the right side of the attack, cut in off the flank and hit a fine strike with his weaker foot into the far corner for what was ultimately the only goal of the game.
Anzhi, in truth, were disappointingly negative at Anfield, rarely breaking forward in numbers until late on in the game as they chased an equaliser, which they didn't deserve. Indeed, only some bizarre refereeing—ruling out a perfectly fine Daniel Agger goal and denying Oussama Assaidi a clear penalty—prevented the scoreline from taking on a more respectable look.
For Liverpool, they will go into the Merseyside derby in good spirits having won their fourth game in the last six matches and having kept three clean sheets in a row.
Here are the player ratings for Liverpool's victorious men on the night.