Kazan has also acted in several Broadway productions.
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 a Greek emigrant from Constantinople.
Kazan was educated at the private Wildwood School, Windward School, and also at the Marlborough School, located in Hancock Park, Los Angeles. She attended Yale University (also her grandfather's alma mater), where she was a member of the Manuscript Society, graduating in 2005 with a Bachelor of Arts degree in theatre.
After her film debut in 2003 playing Samantha in Swordswallowers and Thin Men, Kazan went on to play her first professional stage role in the 2006 off-Broadway revival of The Prime of Miss Jean Brodie starring Cynthia Nixon. In 2007 she had a small role in The Savages, which starred Laura Linney and Philip Seymour Hoffman. She also guest appeared in an episode of the supernatural drama Medium as a friend of the main character's younger self. She next appeared in the films Fracture and In the Valley of Elah. In the fall of the same year, she returned to the stage in a The New Group 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, she appeared on Broadway as Masha in Anton Chekhov's The Seagull opposite Kristin Scott Thomas and Peter Sarsgaard. The same year she had roles 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 appeared 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 movies I Hate Valentine's Day and The Exploding Girl, which were both released in 2009.
In 2010, she had a main role in the comedy-drama happythankyoumoreplease as Mary Catherine, the cousin of Josh Radnor's character. She also starred as Millie Gately in 2010 (alongside Paul Dano, playing her husband) in Kelly Reichardt's independent western drama Meek's Cutoff. In the fall, Kazan played Harper Pitt in Signature Theatre Company's 20th Anniversary production of Tony Kushner's Angels in America. On the small screen, Kazan then appeared in four episodes of HBO's Bored to Death as Nina, the love interest of a fictionalized Jonathan Ames played by Jason Schwartzman.
Her play We Live Here, about a dysfunctional family, received its world premiere production from October 12 to November 6, 2011 at the off-Broadway Manhattan Theater Club in New York City. Among the ensemble cast was Amy Irving and the director was 2010 Obie Award winner Sam Gold.
Her next project, for which she wrote the screenplay, was Ruby Sparks, a comedy-romance film directed by Jonathan Dayton and Valerie Faris, and starring Kazan herself, along with Paul Dano, Chris Messina, Antonio Banderas, Annette Bening, Deborah Ann Woll, and Steve Coogan.
In 2014, her third play Trudy and Max in Love opened at the South Coast Repertory and she played the role of Denise Thibodeau in the HBO miniseries Olive Kitteridge.
In July 2016, it was announced that Kazan had co-written the script for Wildlife, with Paul Dano. The film would be an adaptation of a Richard Ford novel of the same name. Neither Kazan nor Dano will act in the film.
Kazan has been in a relationship with actor Paul Dano since 2007.