AAP reported that Stena named the workers as Peter Meddens from Darwin and Barry Denholm from Lossiemouth in Scotland.
The men were working on the Stena Clyde drill rig in VIC/P43 as part of a six-well drilling campaign for Origin Energy.
"Stena Drilling and Stena Drilling Australia would like once again to offer their heartfelt condolences to their families, friends and colleagues," the company reportedly said in the statement.
"The company and its contractor are in close contact with both families and will continue to offer help and support as best we can at this difficult time.
"Both men will be sorely missed by all at Stena Drilling and Stena Drilling Australia."
The National Offshore Petroleum Safety and Environmental Management Authority said drilling operations remained suspended on the rig while it continued its independent investigation.
"The well remains in a safe condition and continues to be monitored," NOPSEMA said in a statement.
"An electrical fire today in connection with a circuit breaker in the switch control room on the facility caused minor damage and was rapidly extinguished.
"NOPSEMA recognises the distress experienced by crew members, family and friends arising from the fatalities and is committed to conducting an independent, rigorous and thorough investigation into the causes of the incident as efficiently and quickly as possible."