"Traore's transfer is totally in line with the goals that our club has set this season, achieving the highest possible results in the Russian league as well as in Europe," Anzhi general director Aivaz Kaziakhmedov told the club's website.
The highly rated centre-forward becomes Anzhi's first signing in this transfer window.
Traore, 22, joins a host of well-known internationals at the wealthy club from the volatile North Caucasus region, such as Cameroon striker Samuel Eto'o.
Anzhi, bankrolled by billionaire Suleiman Kerimov and coached by former Russia boss Guus Hiddink, finished fifth in the domestic championship last season to qualify for the Europa League.
Traore, who notched 18 goals in 39 games for Kuban last season, helped the club to eighth in the Russian top flight - their best finish to date.
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