Four civilians and an official were killed, and 31 people were injured, interior minister Ihor Klymenko said.
President Volodymyr Zelensky said there were "victims" in the attack, but did not provide a figure.
Rescue efforts were continuing.
Pokrovsk lies about 70km (43 miles) north-west of Donetsk city, which is occupied by Russian forces. Before the war it had a population of around 60,000 people.
Mr Klymenko said the first strike killed four civilians, and that an official from the emergency services was killed in the second strike.
The injured included 19 police officers, five rescuers and a child, he said.
Mr Zelensky said the Russian attacks had struck an "ordinary residential building", publishing a video of a five-storey building that had its top floor destroyed.
Amid scenes of general chaos and confusion it showed civilians clearing away rubble, and rescuers helping people into ambulances.
Pavlo Krylenko, the head of the Donetsk region, said the strikes damaged two "private sector residential buildings, a hotel, catering establishments, shops and administrative buildings".
He warned of the "threat of repeated attacks" and urged residents to take shelter.
Andriy Yermak, head of Ukraine's Presidential Office, said at least two civilians were also killed when Russian guided bombs hit "private houses" in Kharkiv Region on Monday evening.
Five people were injured in those attacks, he said.
The attacks came a day after a Russian "guided bomb" hit a blood transfusion centre in north-eastern Ukraine, killing two people, according to Ukrainian officials.
Ukraine has been trying to regain territory occupied by Russia but has made modest gains since launching a counter-offensive two months ago.
On Sunday Mr Zelensky sought to justify attacks on Russian ships in the Black Sea, telling Argentine newspaper La Nacion that Ukraine "has to find another method to end the blockade of our water".
"If Russia continues to dominate its territory in the Black Sea and blockading, firing missiles, then Ukraine will do the same, which is a fair protection of our chances," he said.
Russia withdrew from a deal allowing the export of Ukrainian grain and warned ships in the Black Sea they could face military action, prompting Ukraine to issue a similar declaration.
Last week, a Russian tanker with 11 crew members was hit by what Moscow said was a Ukrainian attack in the Black Sea. Although Ukraine did not comment publicly, a security service source told the BBC a sea drone had been used.
That followed a similar sea drone attack on a Russian naval ship near the Russian port of Novorossiysk, which is a major hub for Russian exports.
Naval drones, or sea drones, are small, unmanned vessels which operate on or below the water's surface. Research by BBC Verify suggests Ukraine has carried out several attacks with sea drones.