Big-spending Anzhi Makhachkala defeated a second-string Liverpool side 1-0 to move a step closer to a place in the Europa League football knockout stages.
Tall Ivorian Lacina Traore struck the decisive goal in first-half stoppage time, collecting Mbark Boussoufa's neat ball over the top and sending an elegant lob over the onrushing Brad Jones in goal.
Liverpool threatened on occasions, with Jordan Henderson blowing a notable first-half chance, but could not stop Anzhi gaining revenge for their 1-0 loss at Anfield last month.
"I'm quite sure, for the Anzhi fans this is a historical win," midfielder Mbark Boussoufa, adding that Anzhi just wished the scoreline had been more emphatic.
"If we did score the second goal, we would play the last 20 minutes more relaxed," he said.
"At the end the end they created, not big chances, but certainly half-chances. That was a little bit annoying. at the end of the day, we win, and that's the most important thing."
It was Liverpool's first defeat to a Russian team since 1992, when the Reds lost 2-4 to Spartak Moscow.
"It's just one of those things. We'll take it as a team, a disappointment. There's also many positives from tonight," Liverpool coach Brendan Rodgers said.
Anzhi top Group A with seven points ahead of Udinese's game against Young Boys Bern later Thursday, and could secure qualification at home to Udinese later this month, depending on results elsewhere.
"It's a very tight group, it was always going to be tight," Rodgers. "It's a big game for us, the next game against Young Boys."
The defeat extends Liverpool's winless run to four games since beating Anzhi 1-0 at Anfield, and Luis Suarez's record as the only Liverpool player to have scored since then.
Rodgers picked a young side in which veteran center-half Jamie Carragher had made more than twice as many appearances for the club as the rest of squad put together, and left top-scorer Suarez and captain Steven Gerrard, among others, on Merseyside, but rejected claims the decision led to the defeat.
"I don't think it has anything to do with Luis Suarez or Steven Gerrard," he said.
Rodgers had warned that a poor-quality pitch could be a problem for Liverpool, and so it proved as the surface cut up badly, leaving some areas with barely a blade of grass by the end.
Jordan Henderson is under pressure to prove himself at Liverpool but did himself no favours on 23 minutes when he went one-on-one with the goalkeeper off an Adam Morgan through ball and promptly passed to a crowd of Anzhi players, earning Morgan's sarcastic applause.
Anzhi created few clear-cut chances before Traore's opener, although Samuel Eto'o forced a one-handed save from Jones from the edge of the box on 35 minutes.
Joe Cole headed wide just before the break off a free-kick from former England teammate Stewart Downing.
Anzhi goalkeeper Vladimir Gabulov almost gifted Liverpool an equaliser on 77 minutes when he let Suso's long-range shot slip to Joe Cole, but the Englishman fired straight at the goalkeeper.
Traore missed a golden opportunity to make it 2-0 soon after when Eto'o sent him clear of the defence, but the Ivorian dragged his shot wide.
Anzhi played their highest-profile home game to date more than 1,500 kilometers from their home city, as UEFA has ruled it too dangerous to host games in Russia's restive province of Dagestan.