Mr. Gladkov, the Russian governor, initially played down reports of violence, saying there was a “massive informational attack” underway, and sought to calm residents’ nerves in a video posted Monday morning. But by the evening, he said that he was putting the region on a counterterrorism footing, which gives the authorities wide power to establish temporary restrictions on movement, step up identity verifications and control telephone communications.
The Kremlin sought to downplay the incident, with its spokesman Dmitri S. Peskov telling the Tass news agency that it was a Ukrainian attempt to “divert attention from the situation” in Bakhmut, the eastern city that Russian forces claimed over the weekend to have captured after a nearly yearlong battle.
While there have been numerous reports of Ukrainians shelling targets across the border over the course of the 15-month conflict, ground assaults are rare. In early March, the Russian Volunteer Corps claimed it had staged a brief incursion into villages in Bryansk, another Russian region on Ukraine’s border.
The Russian Volunteer Corps is led by a Russian nationalist in exile, and is part of a motley collection of groups of Russian citizens who oppose President Vladimir V. Putin’s rule and have taken up arms for the Ukrainian cause.
Russia has suffered several significant psychological blows during the war, including the explosion that damaged the bridge linking the occupied Crimea peninsula to the Russian mainland and the sinking of the cruiser Moskva, the flagship of the Black Sea fleet. But Andriy Zagorodnyuk, a former Ukrainian defense minister who now advises his government, said this border incursion was a milestone because it involved armed troops, which could force Russia to deploy more of its forces along the border instead of the front lines.
It could also erode Russian unity, he said.
“Russians will see they have problems between their own citizens, so the idea of unified Russia is seriously damaged,” Mr. Zagorodnyuk said.
Oleksandr Chubko, Oleg Matsnev Milana Mazaeva and Riley Mellen contributed reporting.