Anderson, 52, will become CEO effective Sept. 1. With the appointment, Grinstein, 75, will retire from Delta and from its board.
Grinstein had lobbied Delta's board to tap an insider from the company as his replacement. The top internal candidates had been Chief Financial Officer Ed Bastian and Chief Operating Officer James Whitehurst.
Delta said Tuesday that Bastian will add the title president to his duties. The company did not say anything about Whitehurst. In a statement, Delta said Anderson "brings a unique depth of experience to the position, having served in top jobs for several major U.S. corporations."
The board's chairman, Daniel A. Carp, said Anderson "possesses the right blend of seasoned leadership, strategic skills, international experience and airline knowledge the company needs to navigate the industry's challenges and capitalize on its opportunities."
The move could revive speculation about a possible deal between Atlanta-based Delta and Eagan, Minn.-based Northwest .
Anderson has worked at UnitedHealth for nearly three years. Before joining UnitedHealth, he was CEO of Northwest Airlines from 2001-2004. He also serves as a director of Cargill and Medtronic, according to Delta's Web site.