Another 32 people were rescued, the company said on its website, listing the names of 62 workers caught up in the disaster yesterday.
The office of President Ilham Aliyev said one person had died in the blaze that broke out on Friday, with the rescue operation to retrieve the other staff hampered by a violent storm.
The storm had on Friday damaged a gas line on platform number 10 in the deepwater Guneshli section of the Azeri-Chirag-Guneshli offshore oil field, causing the fire, SOCAR said in a statement.
"Despite bad weather conditions, 32 people could be saved," Aliyev said. "According to preliminary information, one person died. At this time, rescue operations continue."
Azerbaijan's state prosecutor's office said the person had died Friday night when the lifeboat he was in with three others was hit by heavy winds.
Staff had taken refuge in lifeboats that had been lowered to 10 metres (35 feet) above the water, it said in a statement.
Prosecutors have opened an inquiry into possible "breaches of fire safety regulations", it added.
President Aliyev signed a decree Saturday announcing the creation of a "government commission led by Prime Minister Artur Rasizade" to coordinate the handling of the incident.
A mainly Muslim country of nine million wedged between Russia and Iran, Azerbaijan is a key partner in projects to deliver Caspian Sea energy reserves to the West through pipelines to Turkey, bypassing Russia.
The Guneshli deposits were discovered in 1981 in the south Caspian Sea, some 90 kilometres east of the Azeri capital Baku.
The deepwater Guneshli section of the project began oil production in 2008. Platform number 10 is operated solely by SOCAR.
There have been a number of fatal incidents on offshore oil rigs in recent years.
In 2011, a drilling platform sank in a storm off Russia's far eastern coast, killing 53 people.
