Russia will hold a national day of mourning on Sunday after the deadly attack unfolded in Crocus City Hall, in western Moscow, on Friday night.
President Putin claimed the suspects were trying to flee to Ukraine, but Kyiv has strongly denied allegations of having any involvement in the shooting.
"They tried to hide and moved towards Ukraine, where, according to preliminary data, a window was prepared for them from the Ukrainian side to cross the state border," he said.
The Russian President vowed to hold those responsible for the "barbaric terrorist act" to account.
"All the perpetrators, organisers and those who ordered this crime will be justly and inevitably punished. Whoever they are, whoever is guiding them," the Russian President said.
"We will identify and punish everyone who stands behind the terrorists, who prepared this atrocity, this strike against Russia, against our people."