Born in Kansas City, Kansas, he started playing soprano sax in his teens before switching to baritone. He joined Bennie Moten's Orchestra in the late 1920s, after leaving high school, and remained with Moten while also performing with Paul Banks and with Jesse Stone's Blues Serenaders. After Moten's death in 1935, he joined Count Basie's band, and remained with Basie until 1950 except for a short period in the US Army. He performed with the Basie band at the From Spirituals to Swing concerts in New York City in 1938 and 1939. Although rarely featured as a soloist, Washington has been described at Allmusic.com as "one of the finest baritonists of his generation" and as the "unsung hero of the baritone-sax".
He later moved to Oklahoma City, where he worked at the city airport. He continued to perform on an occasional basis, and took part in a reunion of the Basie orchestra in 1958. He died in Oklahoma City in 1964 at the age of 54.