Kazan was born in Los Angeles, the daughter of screenwriters Nicholas Kazan and Robin Swicord. Her paternal grandparents were film and theatre director Elia Kazan and playwright Molly Kazan (née Thacher). Elia was an Anatolian Greek emigrant from Istanbul, and Molly was a Mayflower descendant. The family surname was originally Kazantzoglou (Greek: Καζαντζόγλου).
Kazan was educated at the private Wildwood School, Windward School, and at the Marlborough School in Hancock Park, Los Angeles. She attended Yale University, where she was a member of the Manuscript Society, and graduated in 2005 with a Bachelor of Arts in theatre.
