The 26-year-old had been training with the club after being released by Hamburg at the end of last season and did enough to convince coach Gian Piero Gasperini to offer him a contract.
"Genoa CFC can announce the signing of Tomas Rincon," a statement on Genoa's official website said.
"He has been playing for Hamburg SV in recent seasons, and after training this week has passed all subsequent medical tests."
Rincon joined Hamburg from Venezuelan side Deportivo Tachira in 2009 and made over 100 appearances for the Bundesliga outfit.