A man, 69, has been arrested in connection with the shooting at a local Kurdish cultural centre, the prosecutor's department in the French capital said.
The French government asked police to reinforce the protection of Kurdish community sites in the country following the deadly attack.
Also this afternoon, protesters clashed with French police during a demonstration near the shooting scene in the Rue d'Enghien.
The prosecutor's office said the suspect had a prior police record, including an arrest for attacking migrants living in tents a year ago, before adding investigators would examine a "possible racist element" behind the attack.
The man was also wounded "in the face" during in the incident and is being treated in hospital, according to district mayor Alexandra Cordebard.
A lawyer for the Kurdish Community Centre said all three people killed in the shooting were members of the Kurdish community
Suspect is French national and 'acted alone'
French interior minister Gerald Darmanin said the suspect is a French national and known by authorities, but not in the context of political extremism, and was not known to the secret service.
He added the "exact motives" of the attacker are so far unknown, and "as things stand" he acted alone.
The shooting took place at midday local time at a Kurdish cultural centre and a restaurant and hairdresser nearby, district mayor Ms Cordebard said.
President Emmanuel Macron tweeted: "The Kurds of France have been the target of a heinous attack in the heart of Paris.
"Thoughts to the victims, to the people who are struggling to live, to their families and loved ones. Recognition to our law enforcement for their courage and composure."
Paris prosecutor Laure Beccuau said three victims of the shooting had died, while one is in a critical condition and two others are in hospital with less serious injuries.
Police officers cordoned off the area on a busy street with shops and restaurants near the Gare de l'Est train station, and urged people to stay away.
Antiterrorism prosecutors said they are investigating the shooting but have not indicated any sign of a terrorist motive.
Investigation into gunman's motives
One witness told French news agency AFP that seven or eight shots had been fired, sparking chaos in the street.
Shortly after the incident, Mr Darmanin said he would travel to Paris and visit the scene of the "dramatic" shooting, and also confirmed the suspect had been arrested by police.
An investigation into the motives of the gunman continues, Ms Cordebard said, who also told reporters the incident was a "terrible drama".