In a post on Twitter, he wrote that the evacuation was ongoing.
"We are working at all levels of state and local authorities to rescue as many people as possible from the flooded areas," Zelenskky wrote and again blamed Russia for the disaster.
"Russia must be held accountable for this deliberate crime against people, nature, and life itself," he wrote.
Earlier, Ukraine's domestic security service said it had intercepted a telephone call that it said proved a Russian "sabotage group" blew up the dam.
It posted a one-and-a-half-minute audio clip of the alleged conversation on its Telegram channel. DW could not independently verify the recording, and Moscow did not immediately comment on its content.
Russia previously claimed that the dam was destroyed by Ukrainian shelling.
The Nova Kakhovka dam was breached on Tuesday, with both Moscow and Kyiv trading accusations of responsibility for its destruction. Some experts said it might have been due to wartime damage and neglect.
Flooding from the breach extends over 600 square kilometers on the Ukrainian-held side of the Dnieper River as well as the Russian-held side of the river, the governor of the Kherson region earlier said.
Ukrainian Interior Minister Ihor Klymenko said on the Telegram messaging app that four people had died and 13 people were missing in the Kherson region and one person had died in the Mykolayiv region.
A Russian-appointed official said eight people had died in Russian-held territory and more than 5,800 had been evacuated from their homes.
The German Red Cross (DRK), meanwhile, has sent 13.3 tons of disaster aid to assist flood victims. Supplies include water and hygiene kits to address the increased risk of illness due to contaminated water.
"During floods, there is an increased risk of falling ill from contaminated drinking water and a lack of sewage treatment," DRK spokeswoman Rebecca Winkels told the German news agency dpa.
The supplies will be handed over to the Ukrainian Red Cross for distribution.
Here are some of the other developments concerning Russia's war in Ukraine on Friday, June 9:
US announces $2.1 billion weapons aid to Ukraine
The Pentagon announced that it will provide an additional $2.1 billion in long-term weapons aid for Ukraine.
The new assistance package will include funding for more Patriot missile battery munitions, Hawk air defense systems and missiles, and small Puma drones that can be launched by hand.
The latest infusion of funding comes as there are signs that Ukraine is beginning — or about to begin — the much-anticipated counteroffensive to try to take back territories held by Russian forces.
Unlike the US equipment, weapons and ammunition that are more frequently sent from Pentagon stocks and delivered quickly to Ukraine, this money is meant to be spent over the coming months or even years to ensure Ukraine’s future security needs.
Ukrainian MP criticizes lack of strong reaction by international organizations following dam breach
Ukrainian Member of Parliament Oleksiy Goncharenko told DW that an awful catastrophe happened and "we dont see, really, the strong reaction of international organizations."
He echoed statements by President Volodymr Zelenskyy, who criticized the Red Cross and United Nations for what he said was a lack of support on the ground.
"Ukraine controls part of the right bank of the river... the people are evacuated, they receive help, water and also many charities and organizations are doing what they can," he said.
"But on the left bank of the river, which is controlled by Russians, the siutation is catastrophic," he added.
The Kakhovka dam was destroyed earlier this week, with Ukraine and Russia blaming each other for the attack. The dam lies along the Dnieper River and is held by Russian forces.
UN trade official meets Russian deputy foreign minister amid Moscow's threats to abandon Black Sea grain deal
Top UN trade official Rebeca Grynspan met with Russia's Deputy Foreign Minister Sergei Vershinin in Geneva to talk about the Black Sea grain deal.
The agreement allows the safe passage of agricultural products from Ukrainian ports.
Russia has threatened to abandon the deal once again on July 17 if obstacles to its own shipments are not removed.
UN spokesman Stephane Dujarric said "the past months have shown tangible progress" on improving Russian exports.
He added, "Challenges remain but we will spare no effort to overcome all remaining obstacles."
Russian exports of food and fertilizers are not subject to Western sanctions imposed following Russia's war in Ukraine, but Moscow says restrictions on payments and logistics have created a barrier to shipments.
"The United Nations remains resolutely committed to working for global food security by ensuring that essential food and fertilizers reach global markets and are available and affordable for all," Dujarric added.
Estonia bans leader of Russian Orthodox Church
Estonia has banned the leader of the Russian Orthodox Church, Patriarch Kirill, from entering the country.
Kirill joins a new list of 58 people, considered either complicit or involved in human rights violations, no longer allowed into the country, the Estonian Foreign Ministry said.
Kirill last year held a Zoom meeting with Pope Francis and endorsed Russian leader Vladimir Putin's statements for invading Ukraine.
Francis later told an Italian daily during an interview that he told Kirill "the patriarch cannot transform himself into Putin’s altar boy."
Nuclear plant still getting water — IAEA
The breached Kakhovka dam is still supplying Europe's largest nuclear power plant with cooling water, the UN nuclear agency said on Thursday.
"Ukraine's Zaporizhzhia Nuclear Power Plant is continuing to pump cooling water from the Kakhovka reservoir," the International Atomic Energy Agency (IAEA) said in a statement.
It followed claims by the dam's operator to Ukrainian TV, saying the reservoir had gone "below the critical point of 12.7 meters (42 feet)."
If the claims were true, the reservoir would not have been able to supply Zaporizhzhia's ponds, which are needed to cool the plant.
The IAEA said an assessment by its experts indicated that the plant's pumps could likely continue to be operated even at the level of 11 meters or lower.
"In these difficult and challenging circumstances, this is providing some more time before possibly switching to alternative water supplies," IAEA head Rafael Grossi said.
However, Grossi, who is due to visit the plant next week on a third visit since the war started, warned of the "very precarious and potentially dangerous" safety and security situation around the plant, amid the continuing fighting.
Russia says no grounds to extend grain deal — report
Russia's ambassador to Turkey said that while Moscow is continuing consultations with the UN regarding the Black Sea grain deal, there aren't any grounds to extend it, according to a report by the Russian state-owned RIA news agency.
On Wednesday Russia accused Ukraine of an attack on an ammonia pipeline and threatened to end the agreement on shipping grain from Ukrainian ports because of it.
The grain agreement concluded last summer brought an end to a months-long Russian naval blockade of Ukrainian Black Sea ports.
Talks on the agreement are taking place in Geneva.
Russia reports heavy fighting in southern Ukraine
Russia's military leadership has reported heavy fighting in the Zaporizhzhia region in southern Ukraine and claimed to have fought back repeated Ukrainian attempts to break through the front line.
President Vladimir Putin was briefed on the military's claims that a Ukrainian counter-offensive had been repelled.
Kyiv has remained tight-lipped on any mention of an expected counter-offensive and has accused Moscow of spreading lies.
Neither side's battlefield accounts could be independently verified.
Ukraine says one killed in Russia airstrike
Ukraine's interior ministry said one person had been killed, and three were wounded in a Russian airstrike overnight.
Ukraine's military said it shot down four cruise missiles and ten attack drones during the attack.
During an earlier attack, the air force also said two cruise missiles had struck a civilian object in the central Ukrainian region of Cherkasy.
Regional governor, Ihor Taburets, said at least eight people had been wounded in that strike.
Zelenskyy hails Donetsk fighting 'results'
Ukrainian President Volodymyr Zelenskyy has applauded the "results" of heavy fighting in eastern Ukraine's Donetsk region.
Zelenskyy spoke during his daily video message, which he delivered late on Thursday aboard a train after visiting areas affected by the Kakhovka dam breach.
The president acknowledged the heavy fighting in the Donetsk region.
"But there are results, and I am grateful to those who achieved these results. Well done in Bakhmut. Step by step," he said.
US to announce fresh $2 billion arms package — reports
The United States is preparing to announce a fresh package of arms support to Kyiv worth $2 billion (approximately €1.86 billion), Bloomberg News reported late on Thursday, citing administrative officials.
The funds will be presented under the Ukraine Security Assistance Initiative, Bloomberg said.
The package will include a generous amount of air defense munitions. It's also meant to help Ukraine purchase Hawk missile launchers, alongside two types of advanced Patriot air defense missiles.
Biden, Sunak stress 'unwavering support' for Ukraine
US President Joe Biden and British Prime Minister Rishi Sunak have stressed their "unwavering support" for the people of Ukraine in their fight against the Russian invasion.
The two leaders spoke at the White House on Thursday, during Sunak's visit to Biden.
"The UK and the US together with more than 50 partners have committed historic levels of security assistance to Ukraine," Biden said after the talks.
London and Washington are two of the biggest donors to Ukraine, and they play a central role in a long-term, recently announced effort to train Ukrainian pilots on F-16 fighter jets.
rm, kb,rmt/sms (AFP, Reuters)