The fire broke out in the basement of the popular Cuba Libre bar and quickly engulfed the room where some 20 people were celebrating, filling it with thick black smoke and poisonous fumes, a Rouen police officer told Reuters.
"A large number of those who died were killed by the fire, others were overcome by the toxic fumes," the officer said.
One of the injured suffered burns to nearly 90 percent of the body and was in a critical condition, the officer added.
France's interior ministry said a judicial investigation was under way into the fire, which Rouen Mayor Yvon Robert described as "totally accidental".
That allayed initial public concerns that it could have been a terrorist attack in a city that this week buried a priest who was knifed to death by Islamists in his church.
Stunned onlookers stared at the bar's blown out windows and blackened interior on Saturday morning.
"All of a sudden, everything blew," said Valerie Fouquet, who was outside on the terrace of the bar at the time.
"We saw the smoke and we saw the flames, the chairs flew up, the window exploded."
More than 50 firefighters tackled the blaze, the interior ministry said.