Cobb was born in Jefferson County, Georgia, to John A. Cobb and Sarah Rootes Cobb. He was the younger brother of Howell Cobb. He married Marion Lumpkin, who was the daughter of the Supreme Court of Georgia Chief Justice Joseph Henry Lumpkin. Three of their children lived past childhood: Callender (Callie), who married Augustus Longstreet Hull; Sarah A. (Sally), who married Henry Jackson, the son of Henry Rootes Jackson; and Marion (Birdie), who married Michael Hoke Smith. The Lucy Cobb Institute, which he founded, was named for a daughter who died shortly before the school opened. His niece Mildred Lewis Rutherford served the school for over forty years in various capacities.