The winger fired home during the early stages of the second half to earn the hosts an important three points in Group A.
© PA Photos
The first real opening of the half came after 12 minutes as Luis Suarez's effort went straight into the arms of Vladimir Gabulov.
Anzhi's goalkeeper was called into action again soon after as he beat away Glen Johnson's drive.
The hosts were soon in control and Jonjo Shelvey should have done better midway through the half when he blasted over, despite having plenty of time and space to get a shot away.
Liverpool's best opportunity during the opening 45 minutes came when Johnson lost his footing at the crucial time after being put through on goal. The defender claimed that he had been tripped by the Anzhi defender but the referee dismissed the appeals as the sides went in goalless at the break.
It took the Merseysiders eight minutes after the restart to take the lead as Downing was allowed to surge forward before he struck home with his unfavoured right foot.
Suarez nearly doubled their advantage moments later, but he dragged wide.
Sameul Eto'o was allowed to get a rare shot on goal for the visitors following Downing's poor backpass, though he could only fire straight at Brad Jones.
There was a late chance for Anzhi to snatch an equaliser, but they were denied as Martin Skrtl cleared Mehdi Carcela-Gonzalez's effort off the line.