Sevastopol governor Mikhail Razvozhayev said in a video posted on Telegram that three children and two adults had died, and almost 120 people were injured.
The Black Sea port city and naval base on the Crimean Peninsula was annexed by Russia in 2014, but is still internationally recognised as part of Ukraine.
Sevastopol regularly comes under fire from Ukraine, but Sunday’s attack was unusually deadly.
Razvozhayev said the attack hit Uchkuyevka, an area with sandy beaches and hotels.
Videos posted on social media showed people running from the beach as explosions went off and people in swimming outfits carrying a stretcher. Agence France-Presse could not verify their authenticity.
A local news channel on Telegram, ChP Sevastopol, cited witnesses as saying that an elderly woman was killed as she swam in the sea.
The investigative committee, which probes major crimes, said it was opening an investigation into “a terrorist act”.
The governor said Ukraine had launched five missiles which Russian air defences intercepted over the sea, but fragments fell onto the shore, wounding people.
Razvozhayev said shrapnel hit beach areas in the north of the city and set fire to a house and woodland.
A Russian defence ministry statement said Ukraine committed a “terrorist attack on the civilian infrastructure of Sevastopol with US-supplied ATACMS tactical missiles loaded with cluster warheads”.
The ministry said four missiles were downed, but a fifth changed trajectory after being intercepted “with its warhead exploding in the air over the city”.
It added that “all flight missions for US ATACMS are entered by US specialists based on the US’s own satellite reconnaissance data”.
“Such actions will not be left without a response,” the ministry warned.
Ukraine’s military has not commented on the attack, which came a day after a Russian guided bomb strike on the city of Kharkiv hit a residential building, killing two people and injuring more than 50.
On Sunday, Russian strikes hit a house and a children’s educational facility in Kharkiv, killing one person and injuring 10 including two teenagers, regional governor Oleg Synegubov said.
A drone launched by Ukraine on Russia’s southern Belgorod region on Sunday killed a man, the governor said.
Three Ukrainian attack drones struck Graivoron, near the border with Ukraine, said Belgorod governor Vyacheslav Gladkov, with one hitting a car park near a multi-storey block of flats.
“A peaceful civilian was killed. The man died from his wounds at the spot” and three people were wounded, Gladkov wrote on Telegram.
Ukrainian President Volodymyr Zelensky in a social media post urged supporter countries to help Ukraine step up attacks on Russian soil.
“We have enough determination to destroy terrorists on their territory – it is only fair – and we need the same determination from our partners. We can stop Russia,” Zelensky wrote.