Flight UT579, a Boeing 737-800 operated by the Utair airline, was carrying 164 passengers and six crew from Moscow.
Video shows fierce flames from the aircraft after it crashed through the airport fence and fell into a riverbed.
Some of the injured suffered burns, others carbon monoxide poisoning, officials said.
A spokesperson at Sochi airport said one airport employee had suffered a fatal heart attack during the rescue operation.
The jet was attempting to land in strong wind and heavy rain early on Saturday, Russian media reported.
BBC
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