Born in Little Rock, Arkansas, on December 27, 1931, Norris first studied piano at home with his mother, then with John Summers, a local church organist. His first professional performances were with the Howard Williams Band in and around Little Rock during his junior high and high school years. After graduating from high school, Norris played briefly with Mose Allison, then did a two-year tour in the US Air Force. After his time in the Air Force, Norris played with Jimmy Ford in Houston, Texas, then moved to Los Angeles where he became an integral part of the West Coast Jazz scene. While in Los Angeles, he played on Jack Sheldon's first album and on Ornette Coleman's first album, Something Else! The Music of Ornette Coleman (1958) for Contemporary Records.
In 1960, Norris relocated to New York City and formed a trio with guitarist Billy Bean and bassist Hal Gaylor, and the group made one album