Gleeson was born in Sumpter, Oregon. He 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.
- "Death Claims Art Gleeson at Gold Beach". The Bulletin. UPI. November 28, 1964.
- U.S. WWII Draft Cards Young Men, 1940-1947
- Sport Scripts - Los Angeles Times
- "Art Gleeson, 58, Announcer For the Boston Red Sox". The New York Times. November 28, 1964. Retrieved June 20, 2017.