The on-loan Fiorentina player had to be withdrawn after nine minutes of Il Toro's 1-1 draw at Genoa and medical tests have now identified muscle damage as the problem.
A statement on torinofc.it read: "The Argentinian winger has undergone an ultrasound scan which revealed a first-grade lesion to the long adductor muscle of his left thigh.
"Santana will undergo a further ultrasound scan on December 27. The estimated period of his absence is currently one month."
The 30-year-old has made 14 appearances in Serie A, scoring one goal, since moving to the Stadio Olimpico on a season-long loan.
Santana will miss Torino's home game with Chievo on Saturday before a two-week winter break begins.