Enlow has been executive director of the group, started by legendary economist Milton Friedman and his wife, Rose, since late 2004.
St. Angelo, a pioneer in the school-choice movement, had led the foundation since it launched in 1996. He will continue in an advisory role with the title president emeritus and as a member of the board of directors.
"Gordon was the perfect person to lead our Foundation from its infancy, and Robert is the appropriate choice to lead us into the future," Rose Friedman, co-chairwoman of the board, said in a statement.
Before leading the Friedman Foundation, St. Angelo served 22 years as senior program director at Indianapolis-based Lilly Endowment Inc., the largest private charitable foundation in the United States.
Before that, St. Angelo served as co-chairman of Hubert Humphrey's presidential campaign and Democratic Party state chairman for 10 years.