The Tribune de Genève newspaper said the fire at the Tattes centre in the municipality of Vernier was rapidly brought under control.
The 29-year-old victim died from smoke inhalation, the newspaper said in an online report.
The injured were transported to various hospitals in the canton and in the neighbouring canton of Vaud (Nyon and Morges), according to the report.
Emergency services staff responded in force to the blaze, with more than 40 fire fighters and around 35 gendarmes, as well as numerous doctors, nurses and ambulance attendants, the Tribune said.
The fire broke out for reasons not immediately known in the ground floor of ”Building J” in the asylum seekers complex, fire department captain Marc Feuardent told the newspaper.
Thick smoke rapidly filled the two-storey building, causing panic, witnesses said.
Some residents jumped out of windows while others ran in all directions, one resident of the centre told the Tribune.
“Guys were jumping from the third floor, many are injured,” said a 25-year-old man from Mali, identified as Maga.
Medical personnel had to give one of them a cardiac massage on the spot, the man said.
The asylum seekers centre was the site of another fire that left a dozen residents injured in December 2011.
Originally built in 1967 to house seasonal workers, the complex of 12 buildings has been used to accommodate refugees — both families and individuals — since 1994.
Residents at the centre have complained about over-crowded conditions at the centre in the wake of increased numbers of asylum seekers.
According to recent media reports, 660 people were living in facilities designed to accommodate 450.