The Saturday night attack in a lively area of theaters and restaurants near the city's historic opera house was the latest in a series of apparent Islamist strikes in France that have killed some 245 people since 2015.
Panic broke out on the busy Rue Monsigny with people fleeing into bars and restaurants as the man walked along stabbing people, yelling "Allahu akbar" ("God is greatest") before police shot him dead.
Police identified the assailant as Khamzat A., who grew up with his family in Strasbourg, eastern France, a source close to the inquiry told AFP. The city is home to a large community of refugees from the Muslim Russian republic of Chechnya.
He became a French citizen in 2010 after his mother was naturalized, government spokesman Benjamin Griveaux told French television.
Russian news reports said the Russian embassy in Paris was pressing French officials for more information on the knifeman, whose parents have been taken into custody for questioning.
Although Khamzat had no criminal record, he had been on both of France's main watchlists for suspected radicals - the so-called "S file" as well as a more targeted File for the Prevention of Terrorist Radicalisation (FSPRT), which focuses on people judged to be terror threats - since 2016.
One source said he had been questioned by anti-terror investigators last year "because he knew a man who was in contact with a person who had gone to Syria."
Hundreds of Chechens have joined Islamic militant groups in the Middle East, North Africa and elsewhere in recent years.
The Islamic State group said one of its "soldiers" had carried out the Paris attack, according to the SITE monitoring organization, but provided no evidence to back their claim.
The attack again underscored the difficulty in keeping track of suspected extremists by police facing thousands of potential risks, either from homegrown radicals or people who have immigrated to the country.
The FSPRT watchlist alone has nearly 20,000 people, of whom about half are under active surveillance. Thousands of French troops remain on the streets under an anti-terror operation known as Sentinelle, patrolling transport hubs, tourist hotspots and other sensitive sites.