Fourteen others have been rescued so far, though the ongoing search for other survivors has been hampered by freezing temperatures and a driving blizzard.
Survival in the freezing temperatures is reliant on how long the crewmen are in the water, with one Russian official saying that people could stay alive for about six hours before hypothermia set in.
The rig was being towed by a tugboat and icebreaker from the Kamchatka peninsula towards Sakhalin Island when it was caught in the storm.
The crew were waiting to be evacuated by helicopter but the platform capsized and sank before they could board their rescue rafts, the AAP quoted regional ministry spokesman Taimuraz Kasayev as saying.
Russia's Investigative Committee has opened a probe into the accident, which it said might have happened because of a breach of safety regulations or due to the harsh weather conditions.