The Ukrainian army's Operational Command South said their forces killed 11 Russian soldiers and destroyed 11 rocket launchers, three armoured vehicles and a self-propelled howitzer.
Russian-appointed administration chief Vladimir Leontyev confirmed the Ukrainian attacks to Russian state agency Ria Novosti, saying the city of Nova Kakhovka had been shelled four times.
Local Ukrainian officials on Sunday also said Russian artillery hit Ukrainian towns across the river from the Zaporizhzhia nuclear power plant overnight, adding to residents' anguish as reports of shelling around the plant fuelled fears of a radiation disaster.
Russia's defence ministry said there was more Ukrainian shelling of the plant over the past 24 hours, just a day after the two sides traded accusations of targeting Europe's biggest nuclear plant, which has prompted international concern.
Ukrainian nuclear company Energoatom said it had no new information about attacks on the plant.
Captured by Russian troops in March, but still run by Ukrainian staff, the complex on the southern front line of the war has been one of the major hotspots in the six-month-old conflict.
The US State Department said in a statement on Sunday that Russia did not want to acknowledge the grave radiological risk at the Zaporizhzhia plant and had blocked a draft agreement on nuclear non-proliferation because it mentioned such risk.
Regional governor Oleksandr Starukh said on Telegram on Sunday that Russian forces struck residential buildings in the region's main city of Zaporizhzhia, about a two-hour drive from the plant, and the town of Orikhiv further east.
On Saturday, Starukh told Ukrainian television residents were being taught how to use iodine in case of a radiation leak.
Ukraine's military reported shelling of nine more towns in the area on the opposite side of the Dnipro river from the plant in its daily report while the Ria Novosti agency quoted the Russian defence ministry as saying its air force struck a Motor Sich plant in the region where helicopters were repaired.
Reuters could not verify those reports.
Russian Defence Ministry spokesman Igor Konashenkov said nine shells fired by the Ukrainian artillery in two separate attacks landed in the nuclear plant's grounds.
"At present, full-time technical personnel are monitoring the technical condition of the nuclear plant and ensuring its operation. The radiation situation in the area of the nuclear power plant remains normal," he said in a statement.
The United Nations and Ukraine have called for a withdrawal of military equipment and personnel from the plant to ensure it is not a target.
Ukrainian Foreign Minister Dmytro Kuleba said that Russian forces had turned the plant into military base, putting the whole continent at risk, and had no business being there.
"Russian military must get out of the plant," he said on Twitter.
The UN nuclear watchdog IAEA is waiting for clearance for its officials to visit the plant, which its head said on Thursday should be "very, very close".
Ukrainian President Volodymyr Zelenskiy warned on Friday the situation at Zaporizhzhia remained "very risky" a day after it took hours to reconnect two of its reactors to the grid after shelling cut them off.
President Vladimir Putin launched the invasion of Russia's neighbour on February 24, saying a "special operation" was needed to demilitarise the country and remove perceived security threats to Russia.
Ukraine and its allies have dismissed this as a baseless pretext for an imperialist war of conquest that has killed thousands, displaced tens of millions, turned cities to rubble and threatened the global economy with an energy and food supply crisis, sending prices soaring.
Ukraine's foreign ministry said Kuleba would travel to Sweden on Monday followed by a trip to the Czech Republic on Tuesday as part of Ukrainian efforts to cement international support and push for more sanctions pressure on Russia.
In Prague, he will attend an informal meeting of European Union foreign ministers that will discuss an EU-wide visa ban for Russian citizens.
Zelenskiy called for such a ban earlier this month but so far it found support mainly from Estonia, Latvia, Lithuania, Poland and Finland, which all share a border with Russia.
with DPA
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