Ukrainian President Volodymyr Zelensky declared Kherson "ours" after Russia withdrew troops from the city, which the US hailed Saturday as an "extraordinary victory".

"As of now, our defenders are on the outskirts of the city. But special units are already in the city," Zelensky wrote on Telegram, posting footage in which Ukrainian troops appeared to gather with residents.

"We see children running to meet us and greeting us," said Andriy Zholob, the commander of a medical unit currently about 50 kilometres (30 miles) from Kherson.

"We see attractive, smiling faces, flowers, embroidered towels which we display on our vehicles," he added.

Ukraine's parliament published a video of the national anthem being played in a central Kherson square as a small crowd of people, huddled around a bonfire in the dark of night, sang along before the camera zoomed in on a Ukrainian flag flying from a government building.

"The Ukrainian anthem in the center of Kherson," said the caption to the video, published on social media.

In nearby Mykolaiv province, which Russians have failed to capture but have subjected to months of attacks, governor Vitaliy Kim said the entire region, save for the Kinburn cape in the south, was returned to Ukrainian control.

"Now it's official: the entire Mykolaiv region (except Kinburn) has been liberated," Kim wrote on Telegram.

The US hailed Ukraine's "extraordinary victory" in recapturing Kherson from the Russians on Saturday.

"It's a big moment and it's due to the incredible tenacity and skill of the Ukrainians, backed by the relentless and united support of the United States and our allies," US National Security Advisor Jake Sullivan said while travelling to Cambodia with President Joe Biden for a regional summit.

- Ukrainian television back on -

In Kherson, Kyiv's forces reconnected the local television network to Ukrainian broadcasters, after local media reported that retreating Russian forces blew up the television tower and energy facilities, leaving the city without power.

Kyiv's defence ministry said earlier Friday that Kherson "is returning to Ukrainian control and units of the Armed Forces of Ukraine are entering the city."

Ukrainian artillery teams had clear views over Russia's routes of retreat and warned: "Any attempts to oppose the Armed Forces of Ukraine will be stopped."

Russia's defence ministry said "more than 30,000 Russian servicemen, about 5,000 pieces of hardware and military equipment and materiel have been withdrawn".

Kherson was the first major urban hub to fall after President Vladimir Putin ordered Russian troops into Ukraine on February 24.

Its full recapture by Kyiv would be a political and symbolic blow to Putin and open a gateway for Ukraine's forces to the entire Kherson region, with access to both the Black Sea in the west and Sea of Azov in the east.

"Ukraine is gaining another important victory right now and proves that whatever Russia says or does, Ukraine will win," Foreign Minister Dmytro Kuleba wrote on social media.

He posted an amateur video showing Ukrainians removing a billboard near Kherson that proclaimed: "Russia is here forever".

- 'In tears' -

In Ukraine's capital, the news was met with joy.

Wrapped in flags, popping champagne corks and belting out the Ukrainian national anthem, residents of Kherson living in Kyiv gathered in the city's Maidan square to celebrate.

"I didn't believe it at first, I thought it was going to take weeks and months, a few hundred metres at a time, and now we see them arrive in Kherson in one day, it's the best surprise," said Artem Lukiv, 41, a Kherson resident living in Kyiv.

While it would appear a major Russian setback, the Kremlin insisted that Kherson was still part of Russia and that it did not regret annexing the entire Kherson region at a lavish ceremony in late September.

"This is a subject of the Russian Federation. There are no changes in this and there cannot be changes," Kremlin spokesman Dmitry Peskov told reporters.

A full Ukrainian recapture of the Kherson region would disrupt a vital land bridge for Russia between its mainland and the Crimean peninsula, which Moscow annexed from Ukraine in 2014.

- 'Cynical' attack -

Ukrainian officials were initially wary after Moscow announced this week that it would pull forces to defensive positions on the east bank of the river in the city.

Kherson was one of four regions in Ukraine that Putin claimed to have annexed during the September ceremony, vowing at the time to use all available methods to defend it.

Asked by reporters whether Russia regretted annexing Kherson, Peskov said the Kremlin had "no regrets" about the move.

Earlier on Friday, a Russian strike on a residential building in Mykolaiv killed seven people, regional governor Vitaliy Kim said on social media.

An AFP journalist at the scene saw a gaping hole gouged out of a Soviet-style residential building with emergency workers in yellow helmets on site clearing rubble.

Zelensky branded the strike a "cynical response to our successes at the front".