Ukrainian President Volodymyr Zelenskyy said Russian drones were suspected of targeting a nuclear power station early on Wednesday.
The attack in the western Khmelnytskyi region shattered windows at the plant, injuring 20.
The blasts did not affect the plant's operations or its connection to the grid, the International Atomic Energy Agency said, although power was temporarily cut to some off-site radiation monitoring stations.
In his nightly video address, Mr Zelenskyy said the attack provided yet more evidence that tougher sanctions were needed against Russia.
"It is most likely that the target for these drones was the Khmelnytskyi nuclear power station," he said.
"The shock wave from the explosion shattered windows, including on the nuclear power station's premises."
Mr Zelenskyy said every Russian strike, "especially those daring enough to target nuclear power stations and other critical facilities, serves as an argument that pressure on the terrorist state is insufficient".
Regional governor Serhiy Tyurin gave a more complete picture of the extent of the damage, saying more than 1,700 buildings had been damaged.
Mr Tyurin, in a statement posted on Telegram, said these included 282 apartment blocks, more than 1,400 private homes, 41 educational institutions and six healthcare buildings.
IAEA director general Rafael Grossi said: "The fact that numerous windows at the site were destroyed shows just how close it was. Next time, we may not be so fortunate.”
Ukraine's air force destroyed all 11 Russian drones launched overnight, the military said. Damage was caused by blast waves and falling debris, according to the Interior Ministry.
"At night, the enemy struck territory near the Khmelnytskyi nuclear power plant," Ukraine's Energy Ministry said on the Telegram messaging app. "As a result of the explosion, windows in administrative and laboratory buildings have been damaged."
Power lines were also damaged, it said, with more than 1,800 consumers in the adjacent towns of Netishyn and Slavuta facing power cuts.
In Slavuta, 20 buildings were damaged, including one used by a fire and rescue unit and another by a police department, Interior Minister Ihor Klymenko said.
The town's mayor, Vasiliy Sydor, said 20 people were injured.
News agency reporters on the ground were not able to verify the reports.
The Khmelnytskyi region has regularly been a target of attacks by Russian drones and missiles, which analysts link to a military airfield in the region.
Ukraine's five nuclear power stations have been a point of contention in the 20-month-old Russian invasion.
Russia occupied the Zaporizhzhia nuclear plant, Europe's largest such facility, early in the conflict and each side has since routinely accused the other of planning to shell or attack the site.