Dragna was born to Francesco Paolo Dragna and Anna Dragna in Corleone, Sicily, on April 18, 1891. On November 18, 1898, Dragna came to the United States on the S.S. "Alsatia" with his mother, older sister Giuseppa, and older brother Gaetano. They stayed in Brooklyn with Antonio Rizzotto's family, also from Corleone. It is unknown when Dragna's father arrived in the United States. Dragna stayed in New York for ten years before returning to Sicily. As a young man, he joined the Italian Army and later the Sicilian Mafia.