Clean-up and repair operations were under way, and no one was reported hurt in the incident, the agency said.
"At approximately 6:00 am (1100 GMT) responding crews boarded the platform and extinguished the flame," the Coast Guard said.
Meanwhile a 1.4-nautical mile (2.6-kilometer) "rainbow sheen" that pilots had noticed on the water Friday was reported to be dissipating and moving away from a nearby wildlife refuge.
"Spill response equipment and personnel, including boom and skimmers, are on scene as a precaution to help protect the Breton National Wildlife Refuge" the agency said.
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The rig is in the Breton Sound, southeast of New Orleans, near a bank of islands that form the wildlife refuge.
No shore impact or injury to wildlife was reported.
The Coast Guard said that the platform had an estimated 4,000 barrels of crude oil on board in storage tanks.
In 2010 an explosion on the Deepwater Horizon rig further off the Louisiana coast caused millions of barrels of oil to pour into the Gulf.