The 29-year-old Argentine is reportedly set to spend 18 months at Stadio Luigi Ferraris and Genoa have the option of signing him on a permanent basis.
Costa joined Spartak from Valencia in June 2013 and has scored three goals in 35 appearances for the Russian club.
The former Montpellier man will now get his first experience of Italian football for a Genoa side who occupy sixth place in Serie A.
A statement on Genoa's official website said: "Genoa announces that it has acquired from FC Spartak Moscow on a temporary basis with an option to acquire all, the right to benefits of the midfielder Tino Costa."
Costa could make his debut in Genoa's first Serie A game of the year against Atalanta on Tuesday.
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