Gleeson got his start calling games for teams in the California League games during the late 1930s and early 1940s. Gleeson served in the U.S. Navy during World War II and worked for the Armed Forces Network after being discharged from submarine service., From 1946-1949 he was play-by-play announcer for the Oakland Oaks of the Pacific Coast League. In 1950, he joined Mutual Broadcasting System, where he called New York Yankees games with Mel Allen from 1951–1952 and the "Mutual Game of the Day" from 1953-1959. From 1956-1960 Gleeson was the sports director of MBS. Gleeson joined the Boston Red Sox broadcast team in 1960, calling games with Curt Gowdy, Bill Crowley (1960), and Ned Martin (1961–1964) for five seasons. Gleeson died after the 1964 season and was replaced by Mel Parnell in the Red Sox booth.