Born in Smale, Monroe County, Arkansas, United States, Cobbs moved to Chicago in 1951, where he occasionally performed in local clubs with Little Walter, Eddie Boyd and others. He served in the American armed forces and then returned to Chicago, recording a number of singles on such labels as Ruler, a subsidiary of J.O.B. Records.

He first recorded his composition “You Don’t Love Me” in 1960 for Mojo Records, a record label in Memphis, Tennessee, owned by Billy Lee Riley. The recording was leased to Vee-Jay Records for release. Cover versions have been recorded by various artists, including the Allman Brothers Band; Grateful Dead; Richie Kotzen; Kaleidoscope; Quicksilver Messenger Service; John Mayall & the Bluesbreakers; Junior Wells; Magic Sam; and Mike Bloomfield, Al Kooper and Stephen Stills on their 1968 album Super Session. Another cover version was the 1967 rocksteady rendition by Dawn Penn, entitled “You Don’t Love Me (No, No, No)”.

Cobbs later released singles for various labels. He ran nightclubs in Arkansas and Mississippi through the 1970s and 1980s. He went on to release the albums Hey Little Girl for the Wilco label in 1986 and Down to Earth for the Rooster Blues label in 1994.

He has performed at the King Biscuit Blues Festival and the Chicago Blues Festival. He appeared in the 1991 film Mississippi Masala, performing the songs “Angel from Heaven” and “Sad Feelin'”.