Kucka was forced off on the hour-mark at Stadio Sant'Elia in a 2-1 defeat that left Genoa in eighth place in the Italian top flight.
The Slovakia international is likely to be sidelined for at least six months as he prepares to start his rehabilitation.
A club statement read: "Juraj Kucka has undergone diagnostic tests, an MRI showed a rupture of the anterior cruciate ligament in his left knee."
Kucka has made 12 appearances for Genoa this term, scoring twice, but is now unlikely to feature until next season.
His absence will come as a blow to Gian Piero Gasperini, who had led Genoa to five games without defeat before Sunday's loss.
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