"Without a doubt, Russia will itself search for those who committed this crime. They must be found and punished," Kremlin spokesman Dmitry Peskov told reporters.
Russia "will do everything within the framework of international mechanisms to draw attention to this crime," he added.
The defence ministry said last week that the videos showed the "deliberate and methodical murder" of more than 10 servicemen.
The UN said last week it had been made aware of the videos and was looking into them. A report it released earlier last week said there were credible allegations of abuses committed by both sides.
One video shows soldiers apparently surrendering to several military personnel in camouflage and wearing yellow armbands.
The soldiers who are giving themselves up lie down on the ground in the debris-filled backyard of a house. The video abruptly cuts as shots are heard.
Another video filmed from above shows the bodies of around a dozen people surrounded by apparent blood stains.
AFP has not independently confirmed the videos and the Russian defence ministry did not say when the footage was obtained.
Russia's Human Rights Council said the alleged executions took place in Makiivka, a village in the eastern Lugansk region of Ukraine, which the Ukrainian army said it had recaptured last week.