Kazan was born in Los Angeles, the daughter of screenwriter Nicholas Kazan and actress/producer/writer/director Robin Swicord, and the granddaughter of film and theatre director Elia Kazan. Kazan has stated that she did not know her grandfather was famous until she was thirteen. She was educated at the private Marlborough School in Hancock Park and Yale University, from which she graduated in 2005.

Her first professional stage credit was the 2006 off-Broadway revival of The Prime of Miss Jean Brodie starring Cynthia Nixon. She followed this with the Playwrights Horizons production of 100 Saints You Should Know and Jonathan Marc Sherman's Things We Want, directed by Ethan Hawke.

In January 2008, Kazan made her Broadway debut opposite S. Epatha Merkerson and Kevin Anderson in a revival of William Inge's Come Back, Little Sheba. Ben Brantley of The New York Times called her performance "first-rate," adding, "Ms. Kazan is terrific in conveying the character’s self-consciousness." In the fall of the same year, she appeared on Broadway as Masha in Anton Chekhov's The Seagull opposite Kristin Scott Thomas and Peter Sarsgaard.

Kazan was featured in three 2007 films: The Savages, Fracture, and In the Valley of Elah. The following year she appeared in August, Me and Orson Welles and Revolutionary Road.

Kazan is also a playwright. In 2009, her play Absalom premiered at the Humana Festival of New American Plays in Louisville, KY. The play, about a father's tense relationships with his children, had been extensively read and workshopped since Kazan's junior year at Yale University. She capped off the year playing Meryl Streep's daughter in the Nancy Meyers comedy It's Complicated. She is appearing in the Broadway production of A Behanding in Spokane with Christopher Walken and Sam Rockwell until June 6, 2010. She also played a main role in the movie "I Hate Valentines Day" that came out in 2009.

Starting in fall 2010, Kazan will play Harper Pitt in Signature Theatre Company's 20th Anniversary production of Tony Kushner's Angels in America.

Kazan lives in the Carroll Gardens section of Brooklyn.

As of April 2009, Kazan is romantically linked to actor Paul Dano.