Randolph, best known for his 1963 hit "Yakety Sax," was a member of Nashville's "A-Team" of session musicians and played with singers such as Elvis Presley, Eddy Arnold and Roy Orbison, The Tennessean reported.
"The sax is a loud, blare-y instrument," said musician Bob Moore. "But Boots had a way of playing something that would fit with the style and the mood of the song. Whatever he came up with was always outstanding."
Brenda Lee, discussing Randolph's solo on her hit "Rockin' Around the Christmas Tree," said, "I don't care who has recorded that song since, they all copied him."
Randolph worked in nightclubs following a stint in the Army. His break came after he sent a tape of "Yakety Sax" to Chet Atkins, then head of RCA Records in Nashville.
He appeared on the Grand Old Opry, as well as television variety shows.
He recently released "A Whole New Ballgame," an album of standards.
Randolph is survived by his wife Dee, son Randy and daughter Linda.