Born in Los Angeles, California, Kelson was the eldest child of John Joseph Kelson Sr. and Lillian Weinberg Kelson. He started clarinet lessons at age eight, studying with Caughey Roberts. At fifteen, Jefferson High School classmate Chico Hamilton urged him to take up the alto saxophone, and he soon made his professional debut with Jerome Myart. By the time he graduated from Jefferson, he was playing with Hamilton, Buddy Collette, and Charles Mingus at clubs on Central Avenue.
In the 1940s he played with C.L. Burke, Barney Bigard, Marshal Royal, Lucky Thompson, Kid Ory, Benny Carter, Benny Goodman, Lionel Hampton, and Roy Milton. He enlisted in the Navy in October 1942 with Marshal and Ernie Royal, and, after training at Camp Robert Smalls, he was stationed with the Royals with the St. Mary's College Pre-Flight School band.  In the 1950s he also performed with Johnny Otis, Billy Vaughan, Nelson Riddle, Bill Berry, the Capp-Pierce Juggernaut, Ray Anthony, Bob Crosby, and Duke Ellington. He joined Gene Vincent and His Blue Caps in 1958 and is featured on several fine recordings from that period, including Say Mama, She She Little Sheila and Ac-centu-ate the Positive. He worked as a studio musician between 1964 and 1984, in addition to recording with Mercer Ellington and Mink DeVille, touring worldwide with Hampton, Ellington, and Vaughan, and appearing in The Concert for Bangladesh. He semi-retired music in 1984, but returned to performance in 1995 with the Count Basie Orchestra, where he became a regular in 1998. He reverted to his birth name of Kelson that year as well. He died on April 28, 2012, in Beverly Hills, California.
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With Lalo Schifrin
- Gone with the Wave (Colpix, 1964)
- "Central Avenue Sounds: Jackie Kelso". www.calisphere.universityofcalifornia.edu. Retrieved 2012-05-24.
- Marshal Royal, with Claire P. Gordon. Marshal Royal: Jazz Survivor. London: Cassell, 1996
- Leonard Feather and Ira Gitler, The Biographical Encyclopedia of Jazz. Oxford, 1999, p. 380.
- Dik de Heer, Jackie Kelso, at This Is My Story (Biographies), accessed 2012-02-18.
- Central Avenue Sounds: Jazz in Los Angeles (Roth Family Foundation Book in American Music), Clora Bryant et al., ISBN 978-0-520-22098-0