The university announced Monday that Byrne, currently the Wildcats' vice president for athletics, will take over March 1.
The move is pending approval of The University of Alabama System Board of Trustees.
Byrne will replace Bill Battle, who announced Sunday night that he will assume a new role as special assistant to the president.
"We look forward to returning to the SEC and being a part of Alabama's outstanding tradition,'' Byrne said.
Alabama president Stuart R. Bell says Battle will continue to run the department until his successor takes over.
"Bill has done a tremendous job as director of athletics, and has accomplished so much during his career,'' Bell said. "His business expertise, coupled with his coaching experience and his strong understanding of the role an athletic department has in the daily fabric of a university, has allowed us to achieve the great successes we have enjoyed during his tenure. We are blessed to have the continued benefit of his counsel.''
Battle, 75, has run the Alabama athletic department since March 2013. He returned to work on a "limited basis" in July after taking a leave of absence to undergo cancer treatment.
Arizona hired Byrne in 2010 after he served as Mississippi State's athletic director. He is considered one of the top young athletic administrators in the country. He is the son of former longtime Texas A&M athletic director Bill Byrne.
The Wildcats completed construction of the $72 million Lowell-Stevens Football Facility on his watch and finished the first phase of an $80 million renovation of the McKale Center, home to Arizona's basketball program and athletic department.
Battle said in a statement that health was not a factor in his decision to retire, but that he wanted to spend time at his farm in Georgia and in Jackson Hole, Wyoming.
"I want everyone to know that my health was not a factor in this decision as, at this time, I am in full remission and I feel great," Battle said in the statement. "When I came here in 2013, I committed to [former chancellor] Dr. [Robert] Witt that I would be here for four years, or the equivalent of another college degree. Last summer, before my medical procedure, I told Dr. Bell that I was expecting, as were my doctors, to come through that procedure very well, and that I intended to serve out the last year of my contract."
Battle is a former Tennessee head football coach who played on Bear Bryant's first national championship team at Alabama in 1961.
The Associated Press contributed to this report.