The 22-year-old has previously spent two loan spells at Huish Park after struggling to break into the Robins' first team ever since he became a professional with the Championship side in 2009.
Edwards also had loan spells with Bath City and Stockport.
"I am really pleased that Joe has signed for Yeovil Town," manager Gary Johnson told the club's official website.
"It is far better playing first-team football for a lower league team than sitting on the bench and getting no football.
"I believe Joe has a great future and I am very happy that Joe has chosen to further his career at Huish Park."
Edwards will go straight into the squad to play Preston on Saturday.