Johansson was born in New York City on November 22, 1984. Her father, Karsten Johansson, is a Danish-born architect, and her paternal grandfather, Ejner Johansson, was a screenwriter and director. Her mother, Melanie Sloan, a producer, comes from an Ashkenazi Jewish family from the Bronx. Johansson has an older sister, Vanessa, who is an actress; an older brother, Adrian; a twin brother, Hunter (who appeared with her in the film Manny & Lo); and an older half-brother, Christian, from her father's first marriage.
Johansson grew up in a household with "little money", and with a mother who was a "film buff". She and her brother, Hunter, attended P.S. 41 in Greenwich Village in elementary school. Johansson began her theatrical training by attending and graduating from Professional Children's School in Manhattan in 2002.