The Angels said Tony Reagins, the team's director of player development, was promoted to general manager and given a three-year contract.
The Angels were 98-64 this season and won the American League West Division for the third time in four years. They were swept by the Boston Red Sox in the American League Division Series this month.
Stoneman worked in the Montreal Expos organization before joining the Angels. Angels owner Arte Moreno had signed Stoneman to a four-year extension in 2003.
"As time went on I recognized I'm getting older and I really don't have the energy," said Stoneman, 62. "In order to do this job it requires a ton of energy and it really wasn't going to be right to accept to continue."
Stoneman, who helped bring the Angels their only World Series championship, is the fourth general manager to leave his post this season. Atlanta's John Schuerholz and Minnesota's Terry Ryan resigned within the past month, and St. Louis fired Walt Jocketty.