Russia said Ukraine was behind the attack, which it said injured three people. Ukraine has denied involvement.
The giant Russian-held nuclear plant, with six reactors, is on the frontline of the Russia-Ukraine conflict.
The UN’s atomic energy watchdog has repeatedly warned that attacking the plant could risk nuclear disaster.
The International Atomic Energy Agency (IAEA), which has a team of experts at Zaporizhzhia, confirmed “physical impact of drone attacks” at the plant, including at one of the reactors.
The plant’s Russian-installed administration said radiation levels were normal and that there was no serious damage.
The IAEA said the damage had not compromised nuclear safety, but it warned that “this is a serious incident with potential to undermine [the] integrity of the reactor’s containment system”.
IAEA head Rafael Grossi specified there had been “at least three direct hits” against the plant’s “main reactor containment structures”.
“This cannot happen,” he said. “No one can conceivably benefit or get any military or political advantage from attacks against nuclear facilities. This is a no go.”
The Zaporizhzhia plant is the largest in Europe. Russian forces seized it shortly after launching their February 2022 full-scale invasion and have occupied it ever since.
Both Russia and Ukraine regularly accuse each other of shelling the plant and risking a serious nuclear accident.
Mr Grossi has repeatedly warned of the dangers posed to the plant by the conflict.
The plant’s Russian administration has said Ukraine’s armed forces were behind the attack, but Ukraine has denied the allegation.
“Ukraine was not involved in any armed provocations on the site,” Ukrainian Main Intelligence Directorate spokesman Andriy Yusov told the Ukrainska Pravda news website. The plant is “illegally occupied by Russia,” he added.
Mr Yusov accused Russia of endangering the nuclear facility, the civilian population and the environment by carrying out strikes on the plant itself.
Last month the IAEA said its team of experts at the plant had heard explosions every day for a week.
At the time, Mr Grossi said: “For more than two years now, nuclear safety and security in Ukraine has been in constant jeopardy. We remain determined to do everything we can to help minimise the risk of a nuclear accident that could harm people and the environment, not only in Ukraine.”