"Today, the board of directors approved Dzsudzsak's acquisition from Anzhi and signing him to a four-year contract," the club said on their website.
Media reports put the transfer fee for the 25-year-old midfielder at around 15 million euros.
Dzsudzsak became the second player to leave the wealthy club from the volatile North Caucasus region after Brazilian striker Diego Tardelli was sold to Qatar's Al-Gharafa for a reported seven million euros fee last week.
The transfers have reversed a recent trend involving big- spending Anzhi, better known for buying than selling players.
Anzhi, bankrolled by Russian billionaire Suleiman Kerimov, shocked the football world by luring Cameroon striker Samuel Eto'o from Inter Milan on a three-year deal last August.
According to media reports, Eto'o became the highest-paid player in the world with an annual salary of 20 million euros.
Anzhi are cutting back on spending for two reasons, local experts said, the choice of Russian coach Yuri Krasnozhan to lead the club rather than a high-profile foreigner and the league setting a tighter limit on foreign players from next season.
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