The 33-year-old Kelly is coming off surgery to repair a torn ACL but was playing with the Patriots' first-team defense this preseason before he was cut.
The Cardinals were in the market for a tackle after Darnell Dockett went down with a season-ending knee injury.
Kelly played nine seasons with the Oakland Raiders before moving to the Patriots last year. He played in five games for the Patriots before the knee injury ended his season.
Kelly started every game for Oakland his last five years with the Raiders.
To make room for Kelly, the Cardinals released defensive lineman Ryan McBean.