Brendan Rodgers' youthful and inexperienced side put on a valiant show against their Group A opponents but characteristically lacked an end product in the final third
The hosts took a barely deserved lead in the 45th minute when Lacina Traore dinked the ball over his head and past Sebastian Coates before calmly looping an effort over the onrushing Brad Jones for the only goal of the game.
Liverpool manager Brendan Rodgers used the trip to Russia as a chance to rotate his squad, with only Jones and Andre Wisdom surviving from their 1-1 draw against Newcastle on Sunday.
This meant Joe Cole was handed a rare start, while Jordan Henderson, Stewart Downing and Jonjo Shelvey all returned to the side, with youngster Connor Coady handed his first competitive start by the club and Jon Flanagan and Adam Morgan coming into the side.
Anzhi Makhachkala boss Guus Hiddink kept faith with a first XI which included the familiar names of Samuel Eto'o, Yuri Zhirkov and giant striker Traore.
After a bright start by the visitors, restricting Anzhi's movement, they fashioned their first opening in the 23rd minute when Morgan played in Henderson in behind the back four, but instead of electing to shoot, the England international tried to play it across the box and Joao Carlos got back in time to intercept and clear.
The game burst to life in the 35th minute, though, when Zhirkov broke away from Wisdom down the right before finding Eto'o inside the area, who brought a stunning save out of Jones high to his right to keep the scores level.
Anzhi didn't have to wait much longer to take the lead and it came right on the stroke of half-time when Traore flicked the ball over Coates when the ball was fired into him from Mbark Boussoufa before running onto it and lobbing Jones for a superbly well-taken goal just before the break.
The second period followed a similar pattern to the first, with Liverpool sitting deep and looking to break on the counter, but Anzhi almost extended their lead just before the hour mark, with Traore driving just past the post from well outside the area with a powerful effort on his left foot.
They missed a golden opportunity to grab another in the 75th minute when Traore nodded a fantastic ball in from the left by Rasim Tagirbekov straight at Jones, when it looked easier to score than find the goalkeeper.
This prompted an instant response from the visitors, when Vladimir Gabulov spilled a vicious strike from Suso from long range before doing better with Cole's follow-up effort when the midfielder tried to dink it over.
With the game becoming increasingly open, Traore missed another excellent chance shortly afterwards after being played through one-on-one by a long ball over the top, but his arrowed shot just failed to find the bottom far corner.
Despite trying their best to muster some sort of late attack, though, Anzhi were able to hold on comfortably for the win and are now top of Group A with seven points, while the Reds sit just below them on six.