The 26-year-old midfielder spent nearly three years at Vicarage Road, making 29 appearances in the English Championship last season.
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Gian Luca Nani, group technical director, said: "Giorgi has been a top professional at Watford and it feels the right time in his career to allow him to experience playing in one of Europe’s top leagues against some big teams.
"We thank Giorgi for his performances at Watford and look forward to seeing how he develops at Udinese.”