Zinn was born to a Jewish immigrant family in Brooklyn on August 24, 1922. His father, Eddie Zinn, born in Austria-Hungary, immigrated to the U.S. with his brother Samuel before the outbreak of World War I. His mother, Jenny (Rabinowitz) Zinn, emigrated from the Eastern Siberian city of Irkutsk. His parents first became acquainted as workers at the same factory. His father worked as a ditch digger and window cleaner during the Great Depression. His father and mother ran a neighborhood candy store for a brief time, barely getting by. For many years, his father was in the waiters' union and worked as a waiter for weddings and bar mitzvahs.