Minor began his musical career in 1926 playing for a local church orchestra; later he joined the Blue Moon Chasers, a band that was active in and around Dallas. His first major professional work came with Walter Page's Blue Devils, with whom he stayed from 1927 to 1929. He then joined a Texas band called the Blues Syncopaters, led by Ben Smith. In 1930–31 he worked with Earl Dykes, Gene Coy'sqv Black Aces (from Amarillo), Lloyd Hunter's Serenaders, and the Dallas orchestra of Alphonse Trent.qv In 1931 Minor joined the Bennie Moten band.
When Count Basie formed his own band after Moten's death in 1934, Minor became a member of that first Basie unit. He remained with the Count from 1936 to 1941. While with Basie, he was included in the recordings of Gone with What Wind? (1940) and You Can't Run Around (1940), among many others. From 1941 to 1944 he was with the Buddy Johnson band. He also played with Cab Calloway during 1942. In 1945 Minor worked with Mercer Ellington, and at different times he also played and recorded with Lucky Millinder and Willie Bryant. After the 1940s he played freelance. He performed occasionally during the 1960s. Minor was not regarded primarily as a soloist, but rather as a vital section player who rarely took solos. He died in New York on April 11, 1982. ~Handbook of Texas Online