More than 80 firefighters tackled the blaze which broke out during a birthday party taking place in the basement of Cuba Libre in the north-western city in Normandy.
Rouen mayor Yvon Robert said the fire was “totally accidental”, mentioning a possible gas explosion “from the heat in a basement room, doubtless too small for the number of people present”.
It is not known how many people were partying at the Cuba Libre club. The area has been blocked off, and French interior minister Bernard Cazeneuve said a judicial investigation has been opened to discover the cause of the blaze.
One of the six injured people is said to be in a life-threatening condition.
Images from outside the bar on French television showed a large ground-floor window broken open, burned red bar chairs and tattered awning at the front.
The mayor said the fire was “very brief”, adding that the birthday party was “a moment of joy for those involved that ended tragically”.
The local Paris-Normandie newspaper said the fire ignited in the basement of the bar, which was rented for the evening for the birthday celebration.
The paper said the ceiling, which was covered in sound-insulating material, quickly ignited when candles were lit.
The deputy prosecutor at the site later denied initial reports of an explosion. Laurent Labadie said the noise reported by neighbours was the sound of windows exploding from the heat.
The tragedy comes with France on maximum terror alert following two deadly attacks last month.
A priest was killed in his church outside Rouen on July 26, less than two weeks after a Bastille Day attack on the famed promenade in Nice, where a truck mowed down revellers, killing 85 of them. Both attacks were claimed by the Islamic State group.
A funeral Mass for the slain priest was held at Rouen cathedral on Tuesday.