Yelchin was born in Leningrad, USSR (now Saint Petersburg, Russian Federation), the son of Irina Korina and Viktor Yelchin, figure skaters who were national celebrities as stars of the Leningrad Ice Ballet for fifteen years. Yelchin's parents qualified for the 1972 Winter Olympics, but because they were Jewish, were not permitted to participate by the Soviet authorities. Yelchin's family moved to the United States in September of 1989, receiving status as refugees from political and religious oppression. Yelchin's mother now works as a figure skating choreographer and his father as a figure skating coach, having been Sasha Cohen's first trainer. Yelchin's uncle is painter Eugene Yelchin.

Yelchin enjoys playing the guitar, saying that it gives him "a lot of fulfillment", and is a fan of acoustic blues music. He attended the Sherman Oaks Center for Enriched Studies, in Tarzana, California, and enrolled at the University of Southern California in the fall of 2007 to study film.