Green was born in Paris, France, two minutes earlier than her sororal twin sister, Joy (Johanne). She is the daughter of Marlène Jobert, an actress, and Walter Green, a dentist. (Her father Walter has acted in film, appearing in Robert Bresson's film Au Hasard Balthazar.) Her father is of Swedish and French descent, and through him she is the great-granddaughter of the French composer Paul Le Flem. Her mother, a native of French Algeria, later moved to Metropolitan France. Green is Jewish, as is her mother (of Sephardi Jewish heritage). However, she was not raised religious at all; she has said "I feel like a citizen of the world. Life and cinema don't have borders." Green has described her family as "bourgeois", and has said that her sister is very different from her. Green is a natural dark blonde; she has dyed her hair black since she was 15 years old. The French-Swedish actress Marika Green is her aunt. The 1980s pop-star Elsa Lunghini is her first cousin, through their mothers.
Green was raised in France and attended the American University of Paris, an English-speaking institution. She also spent time between Ramsgate, London and Ireland growing up. Green was quiet in school, and developed an interest in Egyptology when she visited the Louvre at age seven. At age 14, after seeing Isabelle Adjani in The Story of Adele H., Green decided to become an actress. Her mother initially feared that an acting career would be too much for her sensitive daughter, but later came to support young Eva's ambitions. Green continued her studies at St. Paul Drama School in Paris, and took an acting course at Webber Douglas Academy of Dramatic Art in London. Around this time, Green returned to Paris, where she performed in several plays. Green stated that in drama school, "I always picked the really evil roles. It's a great way to deal with your everyday emotions."