Stewart Downing's well taken 53rd-minute strike means Liverpool avoided two straight European home losses for the first time in their history after their recent hiccup against Italy's Udinese and they now lead Group A with six points from three matches.
Better still for the Merseysiders, who now lead the Russians and the Italians by two points, Udinese went down to a shock 3-1 loss at unheralded Swiss side Young Boys, throwing away a chance to go top of the section.
Anzhi came to Liverpool boasting former Barcelona and Inter Milan Champions League winner and Cameroon international star Samuel Eto'o in their ranks but he had a quiet night.
"Stewart Downing scored a brilliant goal. He has had a rough ride here, it's been a tough time, but he is a good guy," said Liverpool boss Brendan Rodgers.
"He's an important member of our team and he got a good reception from the Kop."
Also flying the flag for England are Newcastle, who thanked a goal from French winger Gabriel Obertan just after the restart for a second straight home win which keeps the Magpies in charge of Group D.
Alan Pardew's men now lead Bordeaux, already dismissed emphatically at Saint James' Park, by three points after the Girondins - Obertan's former club - could only draw at Portugal's Maritimo.
But there was frustration for another English side, Tottenham, who had to settle for a third straight draw in Group J, held 1-1 at Slovenia's Maribor.
Midfielder Gylfi Sigurdsson rescued a point to keep the Londoners in touch with Group J leaders Lazio, who produced the same scoreline at Panathinaikos.
Spurs trail the Italians by two points after a poor showing at the Stadion Ljudski, where last season Maribor were not good enough to see off second tier English Championship side Birmingham.
Robert Beric fired in from close range for Maribor while Spaniard Iago Falque crossed after the break for Sigurdsson to convert in a goalmouth melee.
Of some consolation to Spurs was Jose Verdu Nicolas's last-gasp leveller for Panathinaikos which denied Lazio a win that looked likely after Georgios Seitaridis had put through his own goal in the first half.
Spurs' boss Andre Villas-Boas insisted afterwards Spurs were still set fair to advance to the last 32.
"The second half was extremely good, the first half was not so good," Villas-Boas opined
"The group is completely open. What we face now is two games at home and I think we are in a good position in the group."
Elsewhere, holders Atletico Madrid thanked goals from Diego Costa and Turkish midfielder Emre Belozoglu for a third straight Group B triumph over Portugal's Coimbra
Turkey's Fenerbache top Group C after a 1-0 success at Cypriot side Limassol while Germany's Borussia Moenchengladbach moved level second with Marseille after defeating the French 2-0.
A surprise in Group F saw Ukraine's Dnipropetrovsk beat Serie A's Napoli 3-1 for a third win on a roll while the Italians are now third behind PSV Eindhoven after the Dutch drew at home to AIK Solna of Sweden.
Former European champions Inter Milan lead Group H after Rodrigo Palacio netted the only goal against Partizan Belgrade but Russia's Rubin Kazan edged Neftci of Azerbaijan by the same score to stay level on points with seven from three games.
French side Lyon thanked a Jimmy Briand goal four minutes from time for a third straight success -- 2-1 over Group I tailenders Athletic Bilbao and Sparta Prague are second after a 3-1 win over Israel's Hapoel Kiryat Shmona.