Lacina Traore's fine individual goal in first-half stoppage-time was enough to send Guus Hiddink's oligarch-backed side to the top of Group A and leave the visitors' hopes of reaching the knockout stage in the balance.
Brendan Rodgers left the likes of Steven Gerrard and Luis Suarez at home with one eye on Sunday's trip to Chelsea but though his inexperienced side toiled keenly, they too rarely threatened Anzhi's goal.
In near-freezing temperatures and in front of a palty crowd - the match played 1,000 miles from Makhachkala for security reasons - it hardly promised to be a classic and there were scant early thrills to reward anyone's journey.
Rasim Tagirbekov found space down the left, only to watch Brad Jones easily claim his weak shot, while 6-foot-8 Traore lingered with intent and almost muscled his way onto a clever through-ball.
A stodgy midfield battle ensued on the patchy turf before Jonjo Shelvey inspired some Liverpool rhythm but promising build-up play fizzled out, John Flanagan dragging a shot well wide, before Jordan Henderson opted for a cut-back that went to waste when the shot was on.
Samuel Eto'o soon had a better chance, his rising strike headed for the top corner until Jones pushed it behind, but Anzhi looked there for the taking until Traore's flash of brilliance.
Mbark Boussoufa's clip forward carried danger but this was about two touches of real quality: a flick over the top of Sebastian Coates and a sidefoot lob over the outrushing Jones.
Rodgers soon sent on the fresh legs of Suso and Dani Pacheco, though departing teenagers Adam Morgan and Conor Coady displayed plenty of endeavour, but Anzhi began to press higher up the pitch and Liverpool struggled to craft real openings.
Traore prowled at the other end but lashed horriby wide of an upright and then headed into Jones' hands from eight yards as Hiddink glowered in the dugout.
Suso had added impetus and forced Vlad Gabulov to parry late on but Joe Cole - disappointing again - saw his shot from an angle blocked and the chance of parity probably undeserved had gone.