Russia's defence ministry said it was only by luck that a radiation accident had been avoided after what it described as an artillery barrage.
Ukrainian President Volodymyr Zelenskiy though said Moscow was responsible and accused it of committing "an open, brazen crime, an act of terror".
In a late night address he demanded sanctions on the entire Russian nuclear industry.
"It is a purely a security issue. Those who create nuclear threats to other nations are certainly not capable of using nuclear technologies safely," he said.
The Russian defence ministry said the generating capacity of one unit had been reduced and power supply to another had been cut. In addition, the nearby city of Enerhodar had power and water supply problems, it said.
"Fortunately, the Ukrainian shells did not hit the oil and fuel facility and the oxygen plant nearby, thus avoiding a larger fire and a possible radiation accident," a ministry statement said.
Enerhodar and the nearby nuclear plant were seized by invading Russian troops in early March, and are still close to the frontline.
U.S. Secretary of State Antony Blinken on Monday accused Moscow of using the plant as a shield for its forces, and Ukraine has accused Russia of shelling its positions from positions near the power station.
"The possible consequences of hitting an operating reactor are equivalent to the use of an atomic bomb," Ukraine's foreign ministry said on Twitter.
Ukraine's state nuclear power company, Energoatom, earlier said the plant was operational and no radioactive discharges had been detected. Two of the six reactors are still operating.
The Russian-installed administration of Enerhodar said on Friday that power lines at the plant had been cut by a Ukrainian artillery strike. The facility continues to be run by its Ukrainian technicians.
Reuters was unable to verify the battlefield reports.
Rafael Grossi, head of the International Atomic Energy Agency, said this week that contact with the plant was "fragile" and communications did not function every day. He appealed for access to determine whether it was a source of danger.
(Reporting by Reuters; writing by Kevin Liffey and David Ljunggren; editing by Sandra Maler and Grant McCool)