A 29-year-old man was killed, and four others were injured.
Paris Mayor Anne Hidalgo also offered her prayers and said she wanted the victims to know "all Parisians are by their side".
The attacker was killed.
The Amaq News Agency said "the person who executed the stabbing in Paris is a soldier of the Islamic State".
The incident took place about 600 yards from the Palais Garnier, the home of the Paris Opera, an area popular with both tourists and Parisians, on a Saturday night when the bars and restaurants there are often full until the early morning hours. When police officers arrived minutes later, he threatened them and was shot to death, according to police union official Yvan Assioma.
Counterterrorism authorities are leading the investigation of the attack.
The attack comes as France is under a constant threat from terrorism. France's military is active in the USA -led coalition against the Islamic State group in Iraq and Syria.
President Emmanuel Macron said France "once again pays the price of blood", but assured the country would yield "not an inch to the enemies of freedom".
The Islamic State group has claimed responsibility for a deadly knife attack carried out in central Paris, according to the SITE monitoring group.
France's military has been active in the coalition since 2014.
The Aamaq statement did not provide evidence for its claim or details on the assailant's identity.
The man attacked five people with a knife, one of whom died, police said. The alleged attacker was killed by police.
The attacker reportedly shouted "Allahu Akbar" (God is greatest) before stabbing his victims.
Paris police officers evacuated people from some buildings in the Right Bank neighbourhood after the attack.
French Interior Minister Gerard Collomb praised the work of police, and is urging people to be careful about spreading fake news, tweeting: "Please do not share misinformation".
Police said the attacker's motives were still unknown.
Four people were wounded during the knife attack in the touristy 2nd arrondissement, or district, of Paris.
- View from Away: The case for Gina Haspel as CIA director
- Worst Mass Shooting In Australia Since 1996: 7 Dead
- NASA, Uber explore urban air mobility
- Imam-ul-Haq ready for Test debut
- Khloe Kardashian shares first look of baby True Thompson Video
- Trump Thanks North Korea For 'Gracious' Pledge To Destroy Nuclear Site
- Giuliani resigns from law firm due to 'pressing demands' of Mueller probe
- Ranger killed, two British nationals kidnapped in Congo national park
- UFC 224 Results: Amanda Nunes, Kelvin Gastelum Claim Victories
- AT&T: Cohen hiring a 'big mistake'