Crawford, known as "Sugar Boy," died early Saturday after a brief illness, The (New Orleans) Times-Picayune reported.
Funeral arrangements were incomplete.
"Jock-A-Mo," which took its lyrics from Mardi Gras Indian chants, was remade by the Dixie Cups as "Iko Iko." Artists such as Dr. John, the Grateful Dead and Cyndi Lauper also recorded variations of "Jock-A-Mo."
Crawford's own career ended for several decades after he was beaten by police in 1963, The Times-Picayune said.
When he was 19, Crawford recorded "Jock-A-Mo." He said the lyrics were "a couple of Indian chants I put together and made a song out of them."
In 1963, while traveling to a show in Monroe, La., with his band, Crawford was stopped by police and badly pistol-whipped. The beating left him with a brain injury and Crawford decided against a comeback because he thought his talent faded, the newspaper said.
Crawford's grandson, pianist and singer Davell Crawford, talked him into performing again. He appeared on his grandson's 1995 album "Let Them Talk," and the two stated performing live, including at the at the New Orleans Jazz and Heritage Festival.
The Crawfords also taped scenes for an episode of the upcoming third season of the HBO drama "Treme."