Denver fired Joseph last month after a 6-10 season. He was 11-21 in two seasons with the Broncos.
The Cardinals hired Kliff Kingsbury as their head coach on Tuesday. Kingsbury replaced Steve Wilks, who was fired after one season (3-13).
Arizona also announced that Bill Davis had been hired as linebackers coach and that Jeff Rodgers had been retained as special-teams coordinator.
Davis, who previously served as Cardinals defensive coordinator in 2009 and 2010, spent the past two seasons as the linebackers coach at Ohio State.