The Ukrainian regional authority confirmed in a Facebook post that Russian forces had captured the plant and said personnel were monitoring the condition of power units to ensure they could operate safely.
Earlier, a video feed from the plant verified by Reuters showed shelling and smoke rising near a five-storey building at the plant compound.
The footage shot at night showed one building aflame, and a volley of incoming shells, before a large candescent ball lit up the sky, exploding beside a car park and sending smoke billowing across the compound.
"Europeans, please wake up. Tell your politicians – Russian troops are shooting at a nuclear power plant in Ukraine," Ukrainian leader Volodymyr Zelenskiy said in a video address.
Zelenskiy said Russian tanks had shot at the nuclear reactor plants, though there was no evidence cited that they had been hit.
The mayor of the nearby town of Energodar about 550 km (342 miles) southeast of Kyiv said fierce fighting and "continuous enemy shelling" had caused casualties in the area, without providing details.
Thousands of people are believed to have been killed or wounded and more than 1 million refugees have fled Ukraine since Russian president Vladimir Putin last Thursday launched the biggest attack on a European state since World War Two.
Early reports of the incident at the power plant sent financial markets in Asia spiralling, with stocks tumbling and oil prices surging further.
"Markets are worried about nuclear fallout. The risk is that there is a miscalculation or over-reaction and the war prolongs," said Vasu Menon, executive director of investment strategy at OCBC Bank.
Russia had already captured the defunct Chernobyl plant north of Kyiv, which spewed radioactive waste over much of Europe when it melted down in 1986. The Zaporizhzhia plant is a different and safer type, analysts said.
Earlier, U.S. President Joe Biden and UK Prime Minister Boris Johnson both spoke with Zelenskiy to get an update on the situation at the plant.
"President Biden joined President Zelenskiy in urging Russia to cease its military activities in the area and allow firefighters and emergency responders to access the site," the White House said.
Johnson said Russian forces must immediately quit their attack and agreed with Zelenskiy that a ceasefire was crucial.
"The prime minister said the reckless actions of President Putin could now directly threaten the safety of all of Europe," Downing Street said.
The head of the International Atomic Energy Agency said he was "deeply concerned" by the situation at the nuclear plant, and that Ukrainian authorities had assured the IAEA that "essential" equipment were unaffected.
Fighting rages, sanctions mount
On Thursday, Russia and Ukraine negotiators agreed to the need for humanitarian corridors to help civilians escape and to deliver medicines and food to the areas where fighting was the fiercest.
Ukrainian presidential adviser Mykhailo Podolyak said a temporary halt to fighting in select locations was also possible.
The negotiators will meet again next week, the Belarusian state news agency Belta quoted Podolyak as saying.
Only one Ukrainian city, the southern port of Kherson, has fallen to Russian forces since the invasion was launched on Feb. 24, but Russian forces continue to surround and attack other cities.
Mariupol, the main port on the Sea of Azov, was surrounded and under heavy bombardment. Water and power was cut off, and officials say they cannot evacuate the wounded.
Video posted on Twitter from Mariupol, and verified by Reuters, showed parked vehicles burning while non-stop firing reverberated around surrounding apartment blocks.
The northeastern city of Kharkiv has been under attack since the start of the invasion, but defenders are holding out in the heavily shelled city.
While no major assault has been launched on Kyiv, the capital has been shelled, and Russian forces unleashed devastating firepower to break resistance in the outlying town of Borodyanka.
In Washington, a U.S. defence official said Russian troops were still 25 km (16 miles) from Kyiv city centre.
The United States and Britain announced sanctions on more Russian oligarchs on Thursday, following on from EU measures, as they ratcheted up the pressure on the Kremlin.
More companies including Alphabet Inc's GOOGL.O Google, footwear giant Nike NKE.N and Swedish home furnishing firm IKEA shut down or reduced operations in Russia as trade restrictions and supply constraints added to political pressure.
Russia calls its actions in Ukraine a "special operation" that is not designed to occupy territory but to topple the democratically elected government, destroy its neighbour's military capabilities and capture what it regards as dangerous nationalists. It denies targeting civilians.
Moscow further tightened the flow of information, restricting access the BBC Russian service and Radio Liberty.
Russian human rights activist and former world chess champion Garry Kasparov called on Western countries to eject Russia from the global police agency Interpol, and impose a no-fly zone over Ukraine.
"Russia should be thrown back into the Stone Age to make sure that the oil and gas industry and any other sensitive industries that are vital for survival of the regime cannot function without Western technological support," Kasparov said.