He was born in Anniston, Alabama. Arnold Caplin, on the liner notes to the album Hot Pianos 1926-1940 reports that Davenport started playing the piano at age 12. His family objected strongly to his musical aspirations and sent him to a theological seminary, where he was expelled for playing ragtime.
Davenport's career began in the 1920s when he joined Banhoof's Traveling Carnival, a medicine show. His first fame came as accompanist to blues musicians Dora Carr and Ivy Smith. He also performed with Tampa Red. He recorded for many record labels, and was a talent scout and artist for Vocalion Records. Davenport suffered a stroke in the early 1930s and lost movement in his hands. He was washing dishes when he was found by the jazz pianist Art Hodes in 1938. Hodes assisted in his rehabilitation and helped him find new recording contracts.
His best-known tune was "Cow Cow Blues". In 1953, "Cow Cow Blues" was re-written by Ahmet Ertegün into Ray Charles' "Mess Around" which was Charles' first step away from his Nat "King" Cole-esque style, and into the style he would employ throughout the 1950s for Atlantic Records.
"Cow Cow Boogie"  was probably named for him, but he did not write it. It was penned by Benny Carter, Gene de Paul and Don Raye. It combined the then popular "Western song" craze (exemplified by Johnny Mercer's "I'm an Old Cowhand") with the big band / boogie-woogie fad. The track was written for the Abbott and Costello film, Ride 'Em Cowboy.
Davenport claimed to have been the composer of "Mama Don't Allow It". He also said he had written the Louis Armstrong hit "I'll be Glad When You're Dead (You Rascal You)", but sold the rights and credit to others.