More than 4,000 people have been evacuated from border areas in Ukraine's northeastern Kharkiv region, the local governor said on Sunday, following a surprise cross-border Russian offensive there that began on Friday.

"In total, 4,073 people have been evacuated," governor Oleg Synegubov wrote on social media, a day after Russian forces claimed the capture of five villages in the region.

Synegubov said that on Sunday a 63-year-old man was killed by artillery fire in the village of Glyboke and a 38-year-old man was wounded in Vovchansk, a border town with some 3,000 residents before the current offensive.

Ukraine on Friday announced that Russia had launched an attack in the Kharkiv region, making small advances into a border zone from where it had been pushed back nearly two years ago.

President Volodymyr Zelensky said on Saturday that Ukrainian troops had been carrying out counterattacks in border villages in Kharkiv region.

"Disrupting Russian offensive plans is now our number one task," he said.

Troops must "return the initiative to Ukraine", the president insisted, again urging allies to speed up arms deliveries.

Ukrainian officials had warned for weeks that Moscow might try to attack its northeastern border regions, pressing its advantage as Ukraine struggles with delays in Western aid and manpower shortages.