But, he said the shooting also fits a wider campaign of intimidation by Ukraine, which has denied any involvement.
Using rhetoric Putin has employed since the beginning of Russia's full-scale invasion of Ukraine, Putin said Russia must find out who benefitted from the attack and who ordered it.
"This atrocity may be just a link in a whole series of attempts by those who have been at war with our country since 2014 by the hands of the neo-Nazi Kyiv regime."
More than 130 people were killed when four men burst into the Crocus City Hall on Friday night, spraying bullets at concert-goers. Islamic State has said it was responsible for the attack. It left at least 180 people injured.
A district court in Moscow on Monday jailed three more suspects - two brothers, Dilovar and Aminchon Islomov, and their father, Isroil Islomov.
They will remain in pre-trial custody until late May.
Dilovar is thought to be the last owner of the car used by suspected attackers to drive to the scene and escape after shooting.
The three men come from Tajikistan, the two brothers are Russian nationals.
Russian authorities had said earlier 11 people had been arrested in connection to the shooting.
Four of them, thought to be the actual shooters, were charged and sent to jail on Sunday, all bearing signs of violence allegedly applied during arrest and interrogation.
France has joined the United States in saying the Islamic State was responsible, with French President Emmanuel Macron saying it would be, quote, "cynical and counterproductive for Russia... to try and turn it against Ukraine."
He also said the same group tried to commit several actions on French soil.
Since the attack, hundreds of Russians have laid flowers outside Crocus City Hall...
And a St. Petersburg concert hall has announced the band Picnic will stage a memorial concert in the city on Wednesday in aid of the victims.
They were the band who had been due to perform Friday night.
