Arians helped guide the Colts to the playoffs this season, stepping in as interim head coach for 12 games (9-3) when Chuck Pagano underwent treatment for leukemia.
Arians has reportedly signed on a four-year deal with an option for a fifth to replace Ken Whisenhunt, who was fired after the Cardinals finished bottom of the NFC West division with a 5-11 record.
The 60-year-old has followed Whisenhunt into a job before, doing so as offensive coordinator at the Pittsburgh Steelers.
This will be his first official NFL head-coaching role, although he spent five seasons in charge of Temple in college football.
Arians becomes the eighth and final appointment of all the teams that fired head coaches at the end of the regular season.
(Reporting by Ben Everill in Los Angeles; Editing by John O'Brien)