At least eight people have been killed and nine others wounded by the Russian shelling, an advisor to the Ukrainian Minister of Internal Affairs has said.
The new development comes after Russia launched massive air and ground assault on the neighbouring country.
The border guard said separately Russian military columns have crossed Ukrainian border into Chernihiv, Kharkiv and Luhansk regions.
Russian forces fired missiles at several cities in Ukraine and landed troops on its coast minutes after President Vladimir Putin authorised what he called a special military operation in the east.
Shortly after Putin spoke in a televised address on Russian state TV, explosions could be heard in the pre-dawn quiet of the Ukrainian capital of Kyiv.
Gunfire rattled near the capital's main airport, the Interfax news agency said, and sirens were heard over the city.
"Putin has just launched a full-scale invasion of Ukraine. Peaceful Ukrainian cities are under strikes," Ukrainian Foreign Minister Dmytro Kuleba said on Twitter.
"This is a war of aggression. Ukraine will defend itself and will win. The world can and must stop Putin. The time to act is now."
U.S. President Joe Biden, reacting to an invasion the United States had been predicting for weeks, said his prayers were with the people of Ukraine "as they suffer an unprovoked and unjustified attack", while promising tough sanctions in response.