Svanidze was the son of Old Bolshevik Alexander Svanidze and Maria Anisimovna (née Korona). His parents were Georgians; his father's family were minor nobility from Kutais Governorate. His mother was from a Jewish family in Tiflis, and was an opera singer at the Tiflis Opera and Ballet Theatre.
Their son was born in Berlin, Weimar Republic, where his father was working as part of a trade mission. They named him after American socialist John Reed, best known for his account of the Russian Revolution, Ten Days That Shook the World. His name was Russified "Dzhonrid", and he was often called "Dzhono", "Dzhoni" or "Vano." He also used the name Ivan, the Russian version of John.
He graduated from Moscow State University with a degree in history. In 1964, he received a Ph.D. from the USSR Academy of Sciences's Institute of African Studies. He received a second doctorate in economics in 1978.