The agency said that as of midday Saturday (0900 GMT) the plant "operates with the risk of violating radiation and fire safety standards".
Russia's defence ministry said Ukrainian forces "shelled the territory of the station three times" from the town of Marganets across the Dnipro River.
In a communique the ministry accused Kyiv of "nuclear terrorism" and said shells landed near areas storing fresh nuclear fuel and radioactive waste.
Radiation levels at the plant "remain normal" the ministry said.
On Tuesday residents in Khortytskyi district -- 45 kilometres (29 miles) north-east from the power plant -- were given iodine pills to reduce radiation risk in case of a leak.
Residents closer to the plant told AFP earlier this month that they received iodine pills at the very start of the war.
On Thursday, the Zaporizhzhia plant was cut off from Ukraine's national power grid for the first time in its four-decade history due to "actions of the invaders", Energoatom said.
Ukrainian president Volodymyr Zelensky said the power cut was caused by Russian shelling of the last active power line linking the plant to the network.
It came back online on Friday afternoon but Zelensky warned "the worst case scenario ... is constantly being provoked by Russian forces".
The International Atomic Energy Agency (IAEA) is urging for a mission to the plant "as soon as possible to help stabilise the nuclear safety and security situation there".
Agency head Rafael Mariano Grossi has said he wants to visit the site within days, warning of potential disaster.
Ukraine energy minister adviser Lana Zerkal said an IAEA inspection "is planned for the next week".
But Zerkal told Ukraine's Radio NV late Thursday she was sceptical the mission would go ahead, despite Moscow's formal agreement, as "they are artificially creating all the conditions so that the mission will not reach the site".
Britain's defence ministry said satellite imagery showed an increased presence of Russian troops at the power plant with armoured personnel carriers deployed within 60 metres (200 feet) of one reactor.
Kyiv suspects Moscow intends to divert power from the Zaporizhzhia plant to the Crimean peninsula, annexed by Russia in 2014. Washington has warned against any such move.
"The electricity that it produces rightly belongs to Ukraine," State Department spokesman Vedant Patel told reporters, saying attempts to redirect power to occupied areas were "unacceptable".
Russia's invasion of Ukraine -- which this week entered its seventh month -- has also prompted a wider global energy crisis.
Anxiety over gas and oil supply has sent prices soaring.
Meanwhile the EU -- which declares itself a staunch ally of Ukraine -- has vowed to wean its 27 member states off Russian energy to protest the nation's attack on its neighbour.
On Friday both Germany and France reported that they expected electricity prices next year to increase tenfold over those this year.
French finance minister Bruno Le Maire pledged on Saturday to keep electricity costs in check.
Paris has put in place an energy price cap to shield households until December 31 and Le Maire said that next year's expected hikes would be "contained increases".
The EU presidency has said it will hold an emergency summit to address the crisis as the winter months approach.
Meanwhile on Saturday, Russia published a decree that makes it easier for Ukrainian citizens to live and work in Russia by scrapping work permits and allowing indefinite residency.
Moscow says 3.6 million Ukrainian nationals, including 587,000 children, have come to Russia since the start of the offensive in late February.