He began his career with Charlie Echols and Paul Howard. In 1932 he joined Duke Ellington. His great technical command of the instrument, with its "creamy tone, neurotic vibrato and range" was featured with Ellington's band every year in compositions such as "Blue Cellophane" and "Golden Cress."
Brown left Ellington's band in 1951 to join a band led by ex-Ellington sideman Johnny Hodges, where he stayed until 1955. After leaving Hodges, Brown took a position for CBS playing studio sessions for five years. Brown rejoined Ellington in 1960 and stayed with him until 1970. After leaving Ellington's band Brown quit performing completely.
Brown fulfilled many roles in the Ellington Orchestra - as a balladeer, technical soloist, and section leader. His highly melodic ballad playing as well as his fast technical style inspired trombonists from Tommy Dorsey to Bill Harris. Trombonist Quentin Jackson said of Brown, "Nobody ever heard a trombone player like that. I never heard nothing [sic] so fast. I never heard a trombone player like that, not like Lawrence was playing in those days." (Dietrich, 74)
Ellington admired Brown's soloing ability, but reserved his highest complements for Brown's manner of playing behind a vocalist, calling Brown an "accompanist par excellence." (Dietrich, 78)
He died in Los Angeles, California.