The Costa Allegra is adrift in the dark more than 200 miles southwest of the Seychelles, near Alphonse Island.
The ship is from the same fleet as the Costa Concordia, which capsized off the Italian coast in January, killing 32.
Costa Cruises said in a statement that the fire broke out in the electric generators’ room. It did not spread and there were no injuries or casualties.
Inspections of the state of the engine room are on-going, the company says.
Tugs and “other naval and aerial units” will reach the ship, which has sent out a distress signal, Costa Cruises says.
Commander Cosimo Nicastro of the Italian coast guard said that it took the crew a few hours to extinguish the fire.
However the ship probably needs to be towed to a Seychelles port, Cosimo Nicastro said.
There are no electric lights on board the ship as the batteries are being used to keep essential machinery going.
The Italian authorities have directed three merchant ships and two fishing vessels towards the stricken liner.
The authorities in the Seychelles say they have sent two tug boats, a coastguard ship and an aircraft to the scene.
The closest vessel to the ship is likely to reach the Costa Allegra at around 2300 GMT, according to AFP.
The first tug boat will not reach the location until 15:00 GMT on Tuesday, the agency reports.
It was due to arrive in the Seychelles on Tuesday.
Further destinations on its itinerary include Alexandria and Naples in the Mediterranean.
Somali pirates are known to operate in the rough area where the ship is adrift, though they have never seized a cruise liner.
A facility on Costa Cruises’ website allowing people to track the Costa Allegra’s position says that “data transmission is temporarily suspended”.
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