The two ships are reportedly part of Russia's "shadow fleet" - a term that refers to the hundreds of tankers used by Russia to bypass Western sanctions imposed after its full-scale invasion of Ukraine in 2022.
The shadow fleet is largely made up of aged tankers, many with obscure ownership or insurance.
The Kairos was struck in the south-west area of the Black Sea and the Virat in a central area further east. Both are on a list of ships subject to sanctions, according to London Stock Exchange data.
The attack employed Sea Baby drones - a type of naval drone produced by Ukraine's security services, known as the SBU.
Turkish authorities said they had been assisting the vessels, and released footage showing two Turkish boats trying to extinguish the fire on the Kairos.
The attacks are seen as a warning by Ukraine, suggesting that ships carrying Russian oil in the Black Sea face the risk of direct attack, not just Western sanctions.
Separately, a major consortium transporting oil from the Caspian region said it had suspended loading in the Russian Black Sea port of Novorossiysk, after an attack by unmanned boats overnight.
The Caspian Pipeline Consortium said the attack had caused significant damage to a mooring point. Russia and Kazakhstan are major stakeholders in the company, which is also part-owned by Western firms including Exxon Mobil, Chevron and Shell.
Kazakhstan's Foreign Minister Aibek Smadiyarov expressed his indignation with the attack, which he said was the third of its kind.
He said Kazakhstan attached particular importance to ensuring a "stable and uninterrupted" energy supply.
The incident had harmed relations between his country and Ukraine, Smadiyarov added, calling on Kyiv "to take concrete measures to prevent the recurrence of such situations".
Meanwhile, Ukrainian President Volodymyr Zelensky said on Saturday that a delegation led by top security official Rustem Umerov was on its way to the US to continue talks on an agreement to end the war.
Umerov is set to lead the Ukrainian delegation as the previous lead negotiator, Zelensky's influential chief of staff Andriy Yermak, resigned after anti-corruption detectives searched his apartment on Friday.
The Ukrainian delegation is expected to meet US officials in Florida on Sunday. Steve Witkoff, US President Donald Trump's overseas envoy, will hold talks in Moscow next week.