Liverpool conceded the top spot in Group A to Europa League opponents Anzhi Makhachkala after losing 1-0 in Moscow on Thursday.
Inter Milan and Lyon, meanwhile, were among the six teams to qualify for the knockout stage but defending champion Atletico Madrid was made to wait after its 16-match winning run in European competition was ended.
Big Azahi striker Lacina Traore scored a brilliant individual goal as the Reds lost focus seconds before halftime.
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.
Anzhi may play 1,000 miles away from their home in Dagestan because of security issues but they remain strong on Russian turf and made it five successive European home wins.
It was always going to be a tough assignment for Liverpool as manager Brendan Rodgers, with Sunday's trip to Chelsea in mind and unperturbed by a weakened team's Capital One Cup exit against Swansea last week, opted to leave his big-name stars at home.
As a result he selected nine Englishmen, in a starting XI with an average age of just under 24, in a 3-5-2 formation.
"Some of the young players were terrific and we are disappointed not to have got something from it," said Rodgers, who left Luis Suarez and Steven Gerrard out of the squad ahead of Sunday's match.
It meant Conor Coady, who captained England Under-17s to their first European Championships title in 2010, made his debut in an icy-cold Lokomotiv Stadium.
The Liverpool-born 19-year-old has progressed through the club's academy and was given his chance having been an unused first-team substitute three times.
Striker Adam Morgan, 18, was making only his second start but, having been prolific throughout youth level Rodgers remains hopeful he will eventually make the step up and he was given the difficult job of leading the line.
At the other end of the scale, Joe Cole, on his 31st birthday, struggled to make an impact again after a woeful 45 minutes against the Swans.
Liverpool and Anzhi can both qualify for the knockout stage from what is proving to be the toughest of all 12 groups by beating Young Boys and Udinese, respectively, at home in two weeks' time.
Reigning champion Atletico Madrid missed a chance to qualify with two games to spare, falling to Academica 2-0 in Coimbra.
Atletico's perfect start to Group B was ended with a defeat that left the holders level on points with Viktoria Plzen -- a 4-0 winner over Hapoel Tel Aviv.
Atletico should also go through in the next matchday as it hosts Hapoel but was sunk on Thursday by a brace from Wilson Eduardo, earning Academica a first European victory since 1969. The visitors were without their star striker Radamel Falcao.
Lyon ensured a top-two finish in Group I by winning 3-2 at 2012 runner-up Athletic Bilbao, which now risks elimination.
Inter Milan joined Rubin Kazan in progressing from Group H with two games to spare by easing past Partizan Belgrade 3-1. Bayer Leverkusen, Metalist Kharkiv and Hannover are also through.
American Alejandro Bedoya scored his first European goal for Helsingborg in a 3-2 loss at Group L leader Hannover. Bedoya, who has been playing in Europe since 2009, tied it at 2 in the 67th minute, and Hannover's Szabolcs Huszti scored on a penalty kick in stoppage time. American right back Steve Cherundolo played for Hannover.
Newcastle stayed top of Group D by drawing 2-2 at Club Brugge while Napoli beat Group F leader Dnipro 4-2 to move into second place.
Fenerbahce and Steaua Bucharest are both five points clear in their groups after a third win in four games but still aren't guaranteed to advance. Fenerbahce beat AEL Limassol 2-0 in Group C, where Marseille drew 2-2 with Borussia Moenchengladbach to leave both teams on five points, and Steaua is in charge of Group E after defeating Molde 2-1 away.
Information from Press Association and The Associated Press was used in this report.