The Costa Allegra is adrift in the dark more than 200 miles southwest of the Seychelles, near Alphonse Island.
Tug boats are on their way to the liner but will not reach it until around 1500 GMT on Tuesday.
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.