Liverpool conceded top spot in Group A to Europa League opponents Anzhi Makhachkala after a 1-0 defeat in Moscow.
Giant striker Lacina Traore scored a brilliant individual goal as the visitors switched off seconds before half-time.
The result, the Reds' first-ever away defeat in the group stage and only their third loss in 22 European matches, allowed the Russian club to assume pole position.
Former Barcelona forward Samuel Eto'o, Anzhi's £330,000-a-week marquee signing, forced Brad Jones into a good save from Yuri Zhirkov's cross after the ex-Chelsea player had robbed Andre Wisdom near Liverpool's right touchline.
Jordan Henderson flicked on a tame near-post header from Stewart Downing's cross and Joe Cole nodded Downing's inswinging free-kick wide before, in added time, Liverpool lapsed.
Traore brilliant spun Sebastian Coates with a left-footed touch and then lobbed Jones with his right for his fourth goal in six matches.
Eto'o unleashed a curling shot which Jones claimed comfortably diving to his left while Traore and Rasim Tagirbekov flashed shots just past the post as the game began to open up.
Cole's last contribution before being replaced by Oussama Assaidi was to force a close-range save from Vladimir Gabulov after he had fumbled Suso's shot.
But Traore should have finished off Liverpool when put clean through by Eto'o but he rolled his effort wide of the upright.
Henderson had a late chance to equalise but ballooned his shot well over.