Joseph “Smokey” Johnson occupied the drum stool in Fats Domino’s band for 28 years and his powerhouse beats can be heard on hundreds of local productions recorded during the ’60s and ’70s by Dave Bartholomew, Wardell Quezergue and Eddie Bo. But it was his own records, such as “Tippin’ Lightly,” “Dirty Red” and 1964’s ingeniously infectious street anthem “It Ain’t My Fault,” that literally foretold the future of funk.

“Wardell and me came up with the melody to...