The incursion was the most serious attack on Russian soil since the beginning of Moscow's offensive in Ukraine, prompting Russian authorities to declare an "anti-terror regime" and evacuate nine border villages.

"In the course of the counter-terrorist operation, the nationalist formations were blocked and destroyed by air strikes and artillery fire," the Russian defence ministry said.

"The remaining (fighters) were driven back to the territory of Ukraine, where they continued to be hit by fire until they were completely eliminated," the ministry said in a statement.

The statement said Russian forces killed more than 70 Ukrainian fighters and destroyed four armoured vehicles.

AFP was unable to independently verify those claims.

The defence ministry blamed Ukrainian authorities for the incursion, saying that "having suffered a defeat in Artemovsk, the Kyiv regime switched to implementing terror attacks against the civilian population."

Moscow used the Soviet-era name for the city of Bakhmut, which Russian forces claimed to have seized over the weekend.

Ukraine has denied involvement in the attack, but members of the anti-Kremlin Freedom of Russia Legion have claimed responsibility for the incursion.