Azimov attacked five people in the French capital, killing one and wounding the others before he was shot dead by police. Terror group Daesh claimed responsibility for the attack.
Investigators working to understand why a 20-year-old French citizen born in Chechnya went on a stabbing rampage in central Paris detained the dead suspect's parents and a friend Sunday, French officials said.
Counterterrorism investigators want to know if the assailant, identified by Chechnya's government as Khamzat Azimov, had help or co-conspirators.
The attacker killed a 29-year-old man and wounded four other people with a knife before police fatally shot him Saturday night.
The suspect was on a police watchlist for radicalism, a judicial official not authorised to speak publicly about the case said.
But he had a clean criminal record and did not know his victims, Interior Ministry spokesman Frederic de Lanouvelle said.
Azimov killed one person and injured four others in a lively neighbourhood near Paris' famed Opera Garnier before he was killed by police on Saturday night.
The Daesh terror group claimed Azimov as one of its "soldiers."
Counterterrorism authorities took charge of the investigation, and President Emmanuel Macron vowed that France would not bow to extremists despite being the target of multiple deadly attacks in recent years.
Paris police officers evacuated people from some buildings in the Right Bank neighbourhood after the attack, which happened on rue Monsigny at about 1900 GMT.
Bar patrons and opera-goers described surprise and confusion in the immediate area.
Beyond the police cordon, however, crowds still filled nearby cafes and the city's night life resumed its normal pace soon after the attack.
Macron: We will not cede
Prosecutor Francois Molins said counterterrorism authorities are leading the investigation on potential charges of murder and attempted murder in connection with terrorist motives.
"At this stage, we have turned this over to the counterterrorist section of the Paris prosecutor's office," Molins told reporters from the scene.
The suspect's parents were detained in Paris' northern 18th district and the friend was detained in the eastern French city of Strasbourg on Sunday afternoon, the judicial official said.
French media reported Sunday that Azimov had lived in Strasbourg, which is 492 kilometers (306 miles) from Paris. It was unclear if he was residing with his parents in the French capital when he carried out the attack.
Witnesses reported hearing the man shouting "Allahu akbar," the Arabic phrase for "God is great," during the attack that happened at about 9 pm in a lively area near the Opera Garnier.
The Daesh group's Aamaq news agency said in a statement early on Sunday that the assailant carried out the attack in response to the group's calls for supporters to target members of the US-led military coalition squeezing the extremists out of Iraq and Syria.
The Aamaq statement did not provide evidence for its claim or details on the assailant's identity.
France's military has been active in the coalition since 2014, and Daesh adherents have killed more than 200 people in France in recent years, including the 130 who died in the coordinated November 2015 attacks in Paris.
President Emmanuel Macron tweeted his praise for police who "neutralised the terrorist" and said "France is once again paying the price of blood but will not cede an inch to enemies of freedom."
The attack occurred near many bars and theaters, as well as the opera.
France's BFM television interviewed an unnamed witness in a restaurant who said a young woman was at the entrance when "a man arrived and attacked her with a knife."
A friend came to her aid and the attacker left, "hitting on all the doors, all the shops," the witness told BFM.
He turned onto another street, and everyone scattered, the witness said.
"I was having a drink with friends and we heard a boom," a witness named Gloria, who had been in a nearby bar, recounted on Saturday night.
She said she went outside to see what happened and "I saw a guy lying on the ground."
Another witness described leaving the opera house and being told to go back inside because of the attack.
Interior Minister Gerard Collomb denounced the "odious attack."