Two people were injured, Regional Governor Oleh Syniehubov said in a video posted on the Telegram messaging app.
"Our energy system came under attack and there was quite serious damage. All crews are at work to eliminate all the negative consequences quickly," Syniehubov said.
"About 80 per cent of the city of Kharkiv and Kharkiv region are without electricity," he added.
The constant threat of further air raids was complicating repair efforts, he said.
Kharkiv Mayor Ihor Terekhov said an "energy site" had been targeted in the city as night-time temperatures dipped to minus 14 degreesC. A city school was badly damaged, he added.
Unofficial Telegram channels posted pictures showing the city, a frequent Russian target located 30km from the Russian border, had been plunged into darkness. Photos showed parts of building interiors seriously damaged.
In Kryvyi Rih, President Volodymyr Zelenskiy's hometown southeast of Kharkiv, Russian drones hit a high-rise apartment building, the head of the industrial city's military administration said.
Earlier, Zelenskiy said Russian strikes on Ukraine's energy sector were "ongoing round-the-clock".
Oleksandr Vilkul said on Telegram that the impact triggered a fire, but residents were safely evacuated.
Reuters could not independently verify the reports and there was no immediate reaction from Russian officials.
The Ukrainian capital Kyiv has been hit by three massive air attacks since the new year, knocking out power and heating to hundreds of buildings.
