The attack took place in a street in the city's Opéra district, an area full of bars, restaurants and theatres which were in full swing on Saturday night.
A waiter at a Korean restaurant, told AFP that a woman came in bleeding and the attacker appeared behind her. He said a young man tried to fend off the assailant who then fled.
"The attacker entered a shopping street, I saw him with a knife in his hand," he said. "He looked crazy."
French Prime Minister Edouard Philippe said police were on the scene "within five minutes" of the attack and that some nine minutes later the assailant was dead, he added. "The speed of the response obviously avoided a heavier toll," he said.
PM Edouard Philippe: The assailant was neutralised 9 minutes after the first call
Dans les 9 minutes après le premier appel, l’assaillant était neutralisé. Nos forces de police ont encore une fois démontré leur remarquable maîtrise et capacité à intervenir. (2/3)Edouard Philippe (@EPhilippePM) 12 mai 2018
One officer tried to restrain the attacker with a taser but when that failed a colleague shot the man dead, a police source told AFP.
Two of those wounded were rushed to hospital in a serious condition but Interior Minister Gérard Collomb later told reporters all the victims were out of danger and would survive their injuries.
The suspect is reported to have been born in the Russian federal republic of Chechnya in 1997.
The parents of the attacker have been taken into custody, the source added.
The Islamic State armed group claimed responsibility, according to the SITE monitoring group, but provided no corroborating proof to back their assertion.
In a message on social media after the attack, France's President Emmanuel Macron said "France once again pays the price of blood but will not give an inch to the enemies of freedom."
Macron: France pays a price in blood once again but does not give an inch to freedom's enemies
La France paye une nouvelle fois le prix du sang mais ne cède pas un pouce aux ennemis de la liberté (2/2).Emmanuel Macron (@EmmanuelMacron) 12 mai 2018
A string of jihadist assaults in the last three years in France have killed more than 245 people.