Brinkley played for the Minnesota Vikings the past four seasons, accumulating a career-high 117 tackles in 2012. He had two sacks and three forced fumbles in 15 starts.
Brinkley, a third-round pick out of South Carolina in 2009, could start alongside Arizona's standout Daryl Washington with the probable departure of Paris Lenon in the team's 3-4 alignment.
Brinkley visited the New York Giants on Wednesday, then came to Arizona on Thursday and decided to sign with the Cardinals.
He is the sixth free agent signing for Arizona, seventh if the re-signing of safety Rashad Johnson is included.