The blaze was sparked by the candles of a birthday cake, which ignited the polystyrene ceiling of the basement room where the celebrations were taking place.
The victims were poisoned by a toxic mix of gases released by the burning plastic, a senior police official told AFP.
“There wasn’t an explosion, it was candles used for a birthday party,” he said.
The blaze started at around midnight at the Cuba Libre bar in the city centre, and the firemen were at the scene by 00:20 am (2220 GMT Friday).
“A fire broke out in a bar in the city centre of Rouen this evening, according to an initial investigation, 13 are dead and six are injured, and more than 50 firemen attended the scene,” Interior Minister Bernard Cazeneuve said in a statement.
One of the injured was in a critical condition, the secretary general of the Seine-Maritime prefecture Yvan Cordier told AFP.
Local official Laurent Labadie who was at the scene of the fire described the blaze as accidental.
Jittery France is on edge after a series of recent attacks by Islamic State-affiliated jihadists, including the killing of a priest in nearby Saint-Etienne-du-Rouvray.
The priest’s funeral was held in Rouen’s cathedral and was attended by more than 2,000 mourners shocked by the brutal killing of the elderly clergyman.