In the incident near Porte de Clichy, the man attacked the three women, aged 19, 24 and 36, with two kitchen knives, severely wounding two of them, Nunez said.
A video aired on Snapchat and verified by Reuters showed a man with long black hair wearing a cream tracksuit and wielding one big knife in each hand and attempting to attack and stab a young woman.
It was unclear if the woman was injured.
"When I saw him stab a woman in the back, I started shouting, 'Come on, let's stop him!' Several people stepped in," Mohamed-Ali Bouhadjar, 26, who filmed the video and posted it on Snapchat, told Reuters.
"One person threw a suitcase at him. He dropped the knives. At that moment, I grabbed him, swept his legs out from under him, and we pinned him to the ground," he added.
Another part of the video shows the same man in a tracksuit lying on the ground immobilised by bystanders while the two knives are seen lying nearby on a pedestrian crossing.
"It is Allah who commanded me," the man lying on the floor says.
Nunez said the man had been detained by an off-duty police officer.
"I pay tribute to him, it was a courageous act," he added.
The motive for the attack remains unknown, Nunez said, adding that police could not verify the identity of the attacker, whose declarations were "incoherent" while he was being arrested.
The French anti-terrorist prosecutor's office said it was weighing whether to open an investigation.
