The ship's lighting, air conditioning and refrigerators were knocked out by the fire, and passengers have been sleeping on deck to escape the sweltering conditions inside their cabins.
"The speed of the ship, despite the hot and humid climate, creates a slight breeze that helps make the situation more comfortable," Costa Cruises said in a statement.
Authorities on Mahe are scrambling to find enough hotel rooms to accommodate the tourists and crew, who are likely to be tired and unwashed after spending four days at sea since the fire broke out.
A Costa Cruises 'care team' has arrived on Mahe to provide assistance to passengers and crew members when the ship finally docks.
"Care Team members immediately met with local authorities and those responsible for the rescue co-ordination in order to define the details of the passengers' arrival on the island and to guarantee the necessary accommodation in local hotels and an efficient organisation for their return back to their final destination," the company said.
Gilbert Faure, the chief executive of the Seychelles' civil aviation authority, said that planes are being lined up to fly passengers back to Rome almost immediately after arriving in port.
He said three aircraft with seats for at least 540 passengers will depart for Rome within hours of the Costa Allegra's arrival at Port Victoria.
The accident came six weeks after another ship owned by the company, the Costa Concordia, capsized off the Tuscan island of Giglio after ramming into a rocky outcrop at night, forcing the panic-stricken evacuation of its 4,200 passengers and crew members.