Kazan stunned defending champion Barcelona 2-1 in the Champions League on Tuesday.
Visiting Lyon handed Liverpool its fourth straight loss in a 2-1 defeat. It was the first time in 22 years that Liverpool has lost four straight games.
Rubin Kazan had earned just one point from the first two matches of its debut appearance in the group stage but twice led at Camp Nou to overturn a team containing Lionel Messi, Zlatan Ibrahimovic and Andres Iniesta.
Barcelona should still advance, but Liverpool may be in trouble after Cesar Fabian Delgado scored in injury time to hand the five-time European champions another loss.
“We need to win at least two more of our games,” Barcelona coach Pep Guardiola said. “It’s a strong group with four champions.”
Liverpool is third in its group, six points behind leading Lyon.
Liverpool’s troubled season continued when it was beaten by Lyon and lost captain Steven Gerrard to a groin injury that could keep him out of Sunday’s Premier League match against rival Manchester United.
The loss heaps more pressure on manager Rafa Benitez, whose team is eighth in the Premier League with four losses in nine games.
“It’s a massive disappointment; a bad result which leaves us a lot to do,” Liverpool defender Jamie Carragher said. “We’re going through a tough patch but we’ve been through them before."
“We had a lot of young lads out there tonight, if we could have hung on, it would have been a good result.”
Liverpool has three points from three Group E matches, six fewer than Lyon and three fewer than Fiorentina, which recovered from conceding a second-minute goal to win 4-3 at Debreceni.
Inter Milan trailed twice at home before the Italian champions salvaged a 2-2 draw with Dynamo Kiev with goals by Dejan Stankovic and Walter Samuel. But Inter is last in Group F after stretching its winless streak in Europe to six matches.
Elsewhere, Fiorentina won 4-3 at Debreceni and Arsenal had a 1-1 draw at AZ Alkmaar.
Sevilla has a five-point lead at the top of Group G with a 3-1 win at Stuttgart, which scored in the 74th minute.
The Spanish team has nine points. Next is Unirea Urziceni, which benefited from two own goals to win 4-1 at Rangers.