Anzhi entered UEFA's competition by finishing fifth in the Russian league last season, after signing Samuel Eto'o and Roberto Carlos, and hiring Guus Hiddink as coach.
It must now find another Russian stadium to host a second qualifying round match on July 19 against Flamurtari Vlore of Albania or Honved of Hungary.
"Such decisions of the UEFA executive committee contradict the values claimed by UEFA," Anzhi general director Aivaz Kaziakhmedov wrote in an open letter to UEFA President Michel Platini published on the club's website. "Anzhi is ready to provide all necessary documentary guarantees from the due state bodies."
An Islamic insurgency which has spread in the North Caucasus region from Chechnya means the team trains near Moscow and flies to Makhachkala for home games.
In May, security forces in Dagestan said they killed warlord Aslan Mamedov and an accomplice in a gunfight in the town of Khasavyurt.
Kaziakhmedov wrote that the club — which is owned by Suleiman Kerimov, a Kremlin-connected billionaire — had invested millions in football in the region "for the good of peace and security."
"After such a decision made by UEFA —what shall we do with all these projects? There is just one question left — maybe we'd better stop this initiative, maybe all this is useless," the Anzhi director wrote.
Platini was urged to "uphold the principles of humanity, philanthropy and good in your activity."
Anzhi also claimed that UEFA enforced the ruling without talking to the club or visiting "to verify whether it's possible or not to hold matches in a city."
UEFA did not immediately respond to requests for a statement on the case.
Anzhi was prevented from hosting home matches on the only previous occasion it qualified for a UEFA club competition.
In the 2001-02 UEFA Cup, a first-round encounter with Scottish club Rangers was played as a single-leg match in neutral Warsaw, Poland, because of security concerns in neighboring Chechnya. Anzhi lost 1-0.
UEFA cleared Anzhi for the Europa League draw despite the club's lavish outlay on transfer fees and wages since "Financial Fair Play" rules took effect last year which seek to limit clubs' spending.
