"When We Leave," which stars Sibel Kekilli as a mother who flees her abusive husband by moving with her 5-year-old son from Istanbul to Berlin, won best narrative feature. Kekilli also won best actress.
The jury that decided the category, which included actors Hope Davis and Aaron Eckhart, said in a statement that "When We Leave" "examines one woman's struggle for personal freedom. It's a theme that is often explored - but rarely told with such humanity, subtlety, craftsmanship or immediacy."
"Monica & David," which won best documentary, chronicles the love between two people with Down syndrome. It was directed by Alexandra Codina, Monica's cousin.
Best actor in a narrative feature film went to Eric Elmosnino, who plays the French musician Serge Gainsbourg in "Gainsbourg, Je t'Aime ... Moi Non Plus."
The festival also singled out two new filmmakers: Kim Chapiron, the director of the youth correctional facility drama "Dog Pound," and Clio Barnard, the director of the British documentary-fiction hybrid "The Arbor."
Jane Rosenthal, who co-founded the festival with Robert De Niro and her husband, entrepreneur Craig Hatkoff, said: "The award-winning films selected by the jury each features strong characters and subjects; these films challenge you to see the world from another perspective."
The awards were handed out Thursday evening at a ceremony in New York. Tribeca, which finishes Sunday, will screen "Freakonomics" on Friday evening as its closing film.