The quarterback, who has previously been a backup player for the Detroit Lions and Indianapolis Colts, will have the chance to compete to be in the starting lineup.
The 28-year-old told reporters: "They need to find an answer at this position. I want to be that answer.
"I'm going to do everything in my power to be that answer, and given that opportunity, I feel like I can."
The move sees Stanton reunited with former Colts offensive coordinator Bruce Arians, who is now the head coach at the Cardinals.