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Having beaten Anzhi 1-0 at Anfield two weeks ago, Liverpool surrendered a two-point lead over their Russian rivals and are now a point adrift after four games.
Chances were few and far between in a lacklustre first-half in which neither side quite managed to gain the upper hand.
Jordan Henderson should have done better when put clean through on the right and found himself one-on-one with the Anzhi goalkeeper Vladimir Gabulov, but he cut the ball back rather than shoot and the danger was cleared.
Cameroon international Samuel Eto'o had a chance to put Anzhi ahead towards the end of the first period as Yuri Zhirkov got forward and crossed from the left, but the first-time Eto'o shot was tipped over at full-stretch by Liverpool goalkeeper Brad Jones.
But then Anzhi broke the deadlock on the stroke of halftime through Traore.
With his back to goal, Traore latched on to a through ball from Mbarak Boussoufa, chipped, turned and lobbed it left-footed over the head of the onrushing Jones from the edge of the area.
Traore could have doubled the lead on the hour when he found himself in space, slipped the ball away from Henderson 20 metres out but fired agonisingly wide.
As the second half wore on Anzhi seemed to grow in confidence and took the game to Liverpool, passing the ball around with relative ease and frustrating the more muscular efforts of the English side.
With 15 minutes to go, Traore had a close range header saved by Jones and a minute later Gabulov was in action at the other end to keep Anzhi in front.
First Pachecho's shot from the edge of the area bounced off his midriff and he had to come smartly off his line to block Joe Cole's close-range shot from the rebound.
Traore then squandered another cast-iron chance, racing clear of the Liverpool defence and sliding his shot wide.
Latest Europa League results on Thursday:
Group A
Anzhi Makhachkala (RUS) 1 (Traore 45) Liverpool (ENG) 0
Udinese (ITA) 2 (Di Natale 47, Fabbrini 83) Young Boys (SUI) 3 (Bobadilla 27, Farnerud 65, Nuzzolo 73)
Group B
Academia Coimbra (POR) 2 (Wilson Eduardo 28, 70 pen) Atletico Madrid (ESP) 0
Viktoria Plzen (CZE) 4 (Kolar 23, 76, Stipek 40, Bakos 84) Hapoel Tel-Aviv (ISR) 0
Group C
Fenerbahce (TUR) 2 (Kuyt 11, M. Sow 41) AEL Limassol (CYP) 0
Marseille (FRA) 2 (Barton 54, A. Ayew 67) Moenchengladbach (GER) 2 (Hanke 20, Arango 90+3)
Group D
Bordeaux (FRA) 1 (Bellion 16) Martimo (POR) 0
Club Brugge (BEL) 2 (Trickovski 14, Jrgensen 19) Newcastle (ENG) 2 (Anita 42, Ameobi 43)
Group E
FC Copenhagen (DEN) 0 VfB Stuttgart (GER) 2 (Ibisevic 76, Harnik 90+2)
Molde (NOR) 1 (Daniel Chima Chukwu 56) Steaua Bucharest (ROM) 2 (Chipciu 21, Latovlevici 37)
Group F
AIK Solna (SWE) 1 (Bangura 12) PSV Eindhoven (NED) 0
Napoli (ITA) 4 (Cavani 7, 77, 88, 90+3) Dnipro Dnipropetrovsk (UKR) 2 (Fedetskiy 34, Zozulya 52)
Group G
Basel (SUI) 1 (Streller 80) Videoton (HUN) 0
Sporting Lisbon (POR) 1 (Ricky van Wolfswinkel 64) Genk (BEL) 1 (Plet 90+1)
Group H
Neftci PFK (AZE) 0 Rubin Kazan (RUS) 1 (Dyadun 16)
Partizan Belgrade (SRB) 1 (Tomic 90+1) Inter Milan (ITA) 3 (Palacio 51, 75, Guarin 87)
Group I
Athletic Bilbao (ESP) 2 (Herrera 48, Aduriz 55-pen) Lyon (FRA) 3 (B. Gomis 22, Gourcuff 45+1, Lacazette 63)
Hapoel Kiryat Shmona (ISR) 1 (Tasevski 3) Sparta Prague (CZE) 1 (Kweuke 24)
Group J
Lazio (ITA) 3 (Kozak 23, 40, Floccari 59) Panathinaikos (GRE) 0
Tottenham (ENG) 3 (Defoe 22, 40, 77) v Maribor (SLO) 1 (Beric 40)
Group K
Bayer Leverkusen (GER) 3 (Hegeler 4, Schrrle 53, M. Friedrich 65) Rapid Vienna (AUT) 0
Metalist Kharkov (UKR) 3 (Taison 4, Xavier 70, Torres 90+3) Rosenborg (NOR) 1 (Dockal 42)
Group L
FC Twente (NED) 0 Levante (ESP) 0
Hannover (GER) 3 (Mame Biram Diouf 3, 49, Huszti 90+1-pen) Helsingborgs (SWE) 2 (Djurdjic 58, Bedoya 67)
