Denver fired Joseph last month after a 6-10 season. He was 11-21 in two seasons with the Broncos.
The Cardinals were looking for a veteran coach who could fill the defensive coordinator's role, general manager Steve Keim said during new coach Kliff Kingsbury's introductory news conference on Wednesday.
Keim also announced Wednesday that the defense would switch back to a 3-4 from the 4-3 they ran under former head coach Steve Wilks. Kingsbury added that he wanted the defense to have an "attacking style."
"We want to be aggressive and get this defense back to where they've been, previous to last year," Kingsbury said.
And under Joseph, that's what Kingsbury will get. The Broncos had the seventh-highest blitz rate at 31.5 percent.
The Cardinals hired Kingsbury as their head coach on Tuesday. Kingsbury replaced 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.