The blaze broke out at around midnight at the basement room of the Cuba Libre bar in the city centre during a birthday celebration.
"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," Cazeneuve said in a statement.
"An inquiry is underway to establish the cause of the fire," he added.
One of the injured was in a critical condition, the secretary general of the Seine-Maritime prefecture Yvan Cordier told AFP.