Downing cracked home an unstoppable effort from outside the box in the 53rd minute after coming in from the left flank to send the Reds top of Group A.
The win was no less than the Reds deserved having been the better side for much of the contest, and they now lead the group with six points from three games.
Liverpool manager Brendan Rodgers fielded a strong side including the likes of Steven Gerrard and Luis Suarez, resisting the temptation to rest a host of players ahead of Sunday’s Premier League derby at Everton and making only three changes from the win over Reading on Saturday.
It was clear Rodgers was not about to underestimate nouveaux riche Anzhi, who lead the Russian top-flight, and are overseen by one-time Chelsea boss Guus Hiddink and had former Barcelona and Inter Milan striker Samuel Eto’o up front.
The Reds were looking to bounce back from a 3-2 home defeat to Udinese in their previous Europa League outing and made a positive start without fashioning anything to really trouble visiting goalkeeper Vladimir Gabulov.
He easily collected a shot from Suarez in the 12th minute and then pushed away an effort by Glen Johnson before Jonjo Shelvey – scorer of three goals in the previous two group games – ballooned an attempt over the bar from the edge of the box.
Eto’o had been relatively quiet, miscontrolling on the occasion when the ball had come to him in a useful position in the area, and Liverpool continued to carve out half-chances, with Oussama Assaidi seeing Gabulov get behind his strike to the near post and then catching his free-kick.
Johnson found himself through on goal in the 35th minute, but failed to get a shot in and took a tumble under the attentions of Kamil Agalarov, with the referee waving the Reds defender’s penalty claims away.
Nuri Sahin headed a corner over and Daniel Agger was next to make a foray forward from defence, trying his luck from distance with an effort that curled high and wide.
Anzhi registered an attempt on goal in first-half stoppage-time when Fedor Smolov fizzed the ball across Brad Jones and wide.
Rodgers added new impetus to his attack at the interval by introducing Raheem Sterling for Johnson, and there were encouraging signs for Liverpool soon after the break, with Martin Skrtel’s powerful drive bringing a save out of Gabulov before Gerrard headed Shelvey’s cross wide.
The Reds kept up the pressure and moments later they made the breakthrough as Downing, who had been shifted to left-back, cut in from out wide and smashed the ball into the net.
Suarez almost swiftly added another with a strike that went just the wrong side of the post, and Shelvey then sent a shot into Gabulov’s arms.
With the lead still only at 1-0, though, Liverpool could not afford to switch off at the back and they survived a scare in the 70th minute when defensive sloppiness saw the ball come to Eto’o, whose effort was saved by Jones.
Assaidi and Suarez went down in the box in separate tussles with the referee only being moved to book the latter for protesting, before Mbark Boussoufa saw a shot deflect wide of Jones’ goal.
Agger thought he had doubled Liverpool’s advantage when he headed the ball from Gabulov’s hand and lashed in, but he also only picked up a booking.
Anzhi issued further warnings, with Skrtel making a goal-line clearance to deny Mehdi Carcela and substitute Lacina Traore striking wide.
Gerrard then fired over and Carcela struck wide at the other end before the full-time whistle confirmed Liverpool’s win.