The film, directed by Drake Doremus, stars Anton Yelchin (The Beaver, Terminator Salvation) and Felicity Jones (Cemetery Junction) as young lovers in a long-distance relationship. Jones also won a Special Jury Award for her performance in the film.
According to The Hollywood Reporter, Paramount and Indian Paintbrush secured a $4m (£2.5m) deal during the festival to release Like Crazy worldwide.
Other big winners included the sci-fi drama Another Earth and Paddy Considine's Tyrannosaur, which picked up two awards each.
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