The incident began just after 11 pm on Tuesday in the downmarket area around Rue Max Dormoy in the 18th arrondissement when the individual pulled out a knife and began randomly stabbing passers-by, media reports said.
Police did not say if the assault might be a terror attack or if was being classed as a drunken rampage by an unstable individual.
The man then made off on foot, after wounding five people who had to be taken to hospital for cuts that were described as not life-threatening.
The attack happened at Rue Marx Dormoy in the north of the capital. Google Maps
A sixth person was also stabbed but the knife did not penetrate his thick winter clothing and he escaped injury.
Parts of the attack were caught on surveillance cameras. Police quickly launched a major operation to track down the attacker.
Images of him were taken from the CCTV footage and distributed to all units in the area, while officers monitoring surveillance camera screens were also able to follow where the suspect was heading.
The man was seen taking refuge in the entrance of a building on Rue Philippe-de-Girard in the 10th arrondissement.
When police finally detained him, they said his clothes were covered in blood and he was clearly drunk, Le Parisien reported.
The knife allegedly used in the attack was found in a nearby street.
The suspect was held overnight in a police cell and was due to be interrogated on Wednesday when he sobered up.
France remains on high alert following a series of knife attacks and attempted knife attacks over the past year. The most serious was a year ago when a soldier guarding the Louvre shot an Egyptian man who tried to attack a security patrol with a machete shouting "Allahu Akbar".
The Egyptian survived and is now awaiting trial for terrorist offences.
Soldiers were deployed on the streets of Paris and other French cities in the wake of a wave of terror attacks by radical Islamists in recent yeas that have killed more than 230 people.