Each of the inductees won more than 2,000 games in managerial careers that spanned more than four decades. Torre became the fifth manager to win at least four World Series titles. La Russa won championships with Oakland and St. Louis. Cox led Atlanta to one championship and 14 straight division titles.
Cox, La Russa and Torre all were present for the announcement, made at the winter meetings. They will be inducted July 27 in Cooperstown, N.Y., along with any players chosen by the Baseball Writers' Association of America.
Greg Maddux and Tom Glavine, 300-game winners who spent a large part of their careers playing for Cox in Atlanta, are among the newcomers to the writers' ballot. Voting runs through Dec. 31, and results will be announced Jan. 8.
Hall chairman Jane Forbes Clark said none of the other nine candidates received more than six votes from the 16-man committee, which met Sunday. Twelve votes, or 75 percent, were needed.
Pioneering players' association head Marvin Miller was on the ballot for the sixth time and New York Yankees owner George Steinbrenner for the second. Miller died last year and Steinbrenner in 2010.
Bobby Cox was born in Tulsa, Oklahoma. He graduated from Selma High School in 1959 and attended Reedley Junior College.
The Associated Press writer Howie Rumberg contributed to this report