SOCAR said on Sunday that one worker had been killed and 33 rescued, out of the 62 who were on the oil rig when the fire started.
"We continue the search-and-rescue operation ... We regard those whom we have not found so far as missing," Khoshbakht Usifzade, SOCAR first vice-president, told a news conference.
Usifzade added that rescuers were also searching for three more workers who had been swept into the sea in an accident on another oil platform on Friday.
SOCAR's other vice-president said that the search operation would be continue through the night.
"Despite all the efforts, regrettably, no one has been found," Khalig Mamedov told a news conference.
"This is the biggest tragedy in SOCAR's history," he said.
Mamedov said that rescuers had found several life vests and fragments of a boat, adding that one gas well was still on fire.
The search-and-rescue operation involved seven vessels and four helicopters of the state border service, as well as four Emergency Ministry helicopters and the Azerbaijan Caspian Shipping Company's ships.
The fire started on Friday after the storm caused the partial collapse of one of the facilities on the platform, damaging a natural gas pipeline.
Daily production was 920 tonnes of oil and 1.08 million cubic meters of gas.
About 60 percent of SOCAR's oil production passes through this platform, meaning the state company's output will be temporarily hit.
The bulk of Azerbaijan's oil is produced elsewhere, however, including on fields operated by British oil major BP.
SOCAR will review safety measures on its platforms, many of which were built in Soviet times.