Power was born in Cincinnati, Ohio, in 1914, son of Helen Emma "Patia" (née Reaume) and the English-born American stage and screen actor Tyrone Power, Sr., often known by his first name 'Fred'. Power was descended from a long theatrical line going back to his great-grandfather, the actor and comedian Tyrone Power (1795–1841). Tyrone Power's sister, Ann Power was born in 1915, after the family moved to California. His father's ancestry included Irish, English, Scottish, Italian, German, and French Huguenots (the latter through his paternal grandmother's Lavenu and Blossett ancestors). His mother was Roman Catholic, and her ancestry included the French-Canadian Reaume family and Germans from Alsace-Lorraine.
Through his paternal great-grandmother, Anne Gilbert, Power was related to the actor Laurence Olivier; through his paternal grandmother, stage actress Ethel Lavenu, he was related by marriage to author Evelyn Waugh; and through his father's first cousin, Norah Emily Gorman Power, he was related to the theatrical director Sir (William) Tyrone Guthrie, founder of the Stratford Festival (now the Stratford Shakespeare Festival) in Canada and the Tyrone Guthrie Theatre in Minneapolis, Minnesota.