"We would like to say special thanks to the Lokomotiv Football Club for providing their home arena for our match," Anzhi said on their website.
"We also want to thank the Moscow police for providing a high level of security during the match."
Thursday's match, where Anzhi beat Dutch club AZ Alkmaar 1-0, was attended by several hundred Moscow fans, who racially abused Anzhi players and threw objects on to the pitch.
Some fans were pictured giving a Nazi salute and chanting slogans such as "Russia is (only) for Russians" and "Caucasians go home".
The fans said they were protesting against the club from the volatile North Caucasus region playing their games in Moscow.
UEFA has banned Anzhi from playing at home in the violence-plagued Dagestan region this season for security reasons.
Anzhi coach Guus Hiddink criticised the fans' behaviour.
"I'm not sure if those were the real football fans," the Dutchman told reporters after the match.
"I think it was a provocation against our club. I don't know what UEFA will do next. They may ask us to play elsewhere."
Anzhi striker Shamil Lakhiyalov was even more forthright, saying: "I think it would be better if we play our home games in Europe instead of Moscow."
The head of the violence-plagued Dagestan region on Friday appealed to the Russian people to fight nationalism and racism.
"While most Russian fans cheered Anzhi's victory, others have tried to use the match to incite nationalistic hysteria and hatred between various ethnic groups in Russia," Dagestan leader Magomedsalam Magomedov said in a statement.
Thursday's crowd trouble was the latest in a series of fan violence in Russian football.
Also on Thursday, police in St Petersburg opened a criminal case after several Anzhi supporters had been attacked by local fans in Russia's northern capital.
Last week, several Zenit St Petersburg supporters said they were badly beaten by police and stadium security staff after a league game against Anzhi in Makhachkala.
One of the fans, Alexander Makasin, was on Friday admitted to an intensive care unit in St Petersburg with damage to his spleen and other internal injuries, the fans' group said on their official website.
In March, a fan threw a banana at Anzhi's Congolese defender Christopher Samba at a league game at Lokomotiv Stadium.