The 54-year-old will step down at the end of October, following a decade at Emirates Stadium, and take up the same role at Milan, who were recently taken over by American company Elliott Management, on December 1.
Raul Sanllehi and Vinai Venkatesham will head up football operations at Arsenal, taking up the head of football and managing director roles respectively.
Sanllehi, a former director of football at Barcelona, joined the Gunners as head of football relations in February, while Venkatesham is currently the club's chief commercial officer.
Gazidis headed up the interview process that led to Unai Emery's appointment as head coach, succeeding long-serving manager Arsene Wenger.
His overhaul of Arsenal's structure, including the appointments of Sanllehi and head of recruitment Sven Mislintat, arguably helped to herald the end of the Wenger era as a degree of power was wrestled away from the veteran Frenchman, and Gazidis has described his pending departure as "the hardest decision of my life".
"For the last 10 years I have been privileged to dedicate myself to this great club. Arsenal is entering a new chapter and I have done everything I can to ensure that it is strongly placed to take on that challenge," he said in a statement on Arsenal's official website.
"This includes world-class facilities and outstanding leaders in every sector who carry the values of the club, including, of course, Unai Emery, Raul Sanllehi and Vinai Venkatesham, in whom I have enormous faith.
Raul Sanllehi and Vinai Venkatesham appointed head of football and managing director respectively, as Ivan Gazidis leaves for A.C. Milan— Arsenal FC (@Arsenal) September 18, 2018
"Although it is very hard to do - the hardest decision of my life - I believe that, after 10 years, it is the right time for me to step aside to allow new leadership, energy and ideas to take the club forward into this exciting new era.
"While this is the most difficult and challenging course for me, I am excited to see what the future holds for this great club.
"After so many years at Major League Soccer and Arsenal, I am now looking forward to joining one of the world’s other great clubs, AC Milan, and working to restore it to its rightful place in football. Until then, I will continue to devote absolutely all my energy until my last day to ensure an orderly transition for the benefit of Arsenal Football Club."
Meanwhile, chairman Chips Keswick believes the club are in a stronger position thanks to Gazidis, but is confident about the change in direction.
"Ivan has led Arsenal with skill and dedication and leaves us in a strong position," he said. "The club has been transformed and he has recruited talented people, such as Raul, Vinai and of course Unai Emery, who will take us to future success.”