The 32-year-old, who enjoyed 10 years with the Robins, spent the second half of last season on loan at Wolves.
His move to Molineux helped ease the pressure on Cheltenham's wage bill although he failed to make a first-team appearance for the Championship title winners.
Grayson told Leeds' official website: "Shane is a very experienced goalkeeper and he ended last season on loan at Wolves, who held him in very high regard.
"He's desperate to make an impression and has the hunger and the desire to succeed at this football club. He is another player who we've signed, and are looking to sign, who has the experience of this division and the know-how to help us achieve our goal of promotion."
Higgs will provide healthy competition for Casper Ankergren and Alan Martin.