The rig was being towed with 67 crew members aboard about 200 kilometres off the coast of Sakhalin Island when it encountered stormy conditions.
The Russian Emergencies Ministry said the rig began to sink after a large wave smashed through portholes in the crew's dining room.
A five-metre high wave washed away the rig's lifeboats, leaving the crew with no escape. It sank several hours later.
14 people were pulled alive from the frigid water, but freezing temperatures, harsh winds and a strong blizzard hampered efforts to find the 49 others.
Helicopters and airplanes patrolled the area Sunday, but called off their search just after sunset. Officials say that two boats will continue the search overnight, and the air search will resume Monday morning.
The surface temperature of the Sea of Okhotsk at this time of year ranges from 4°C to -2°C. Hypothermia sets in quickly in water 8°C and colder.
Dmitry Dmitriyenko, governor of the Murmansk region in Russia's northwest -- where 33 of the men come from -- urged friends and families to stay hopeful, but admitted the likelihood of the men surviving in the frigid water is slim.
Officials are investigating the incident, hoping to find the exact cause.
With files from the Associated Press