Pavlo Kyrylenko, governor of the Donetsk region, said about three dozen people could be trapped in the rubble. Rescuers have made contact with two people who are under the wreckage.
Mr Kyrylenko said the town of about 12,000 people was hit by Uragan rockets, which are fired from truck-borne systems.
Chasiv Yar is about 12 miles south-east of Kramatorsk, a city that is expected to be a major target of Russian forces as they grind west.
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The Donetsk region is one of two provinces along with Luhansk that make up the Donbas region, where separatist rebels have fought Ukrainian forces since 2014.
Last week, Russia captured the city of Lysychansk, the last major stronghold of Ukrainian resistance in Luhansk.
Russian forces are raising “true hell” in the Donbas, despite assessments they were taking an operational pause, Luhansk governor Serhiy Haidai said on Saturday.
After the seizure of Lysychansk, some analysts predicted Moscow’s troops would take some time to rearm and regroup.
But Mr Haidai said: “So far there has been no operational pause announced by the enemy. He is still attacking and shelling our lands with the same intensity as before.”
He later said the Russian bombardment of Luhansk was suspended because Ukrainian forces had destroyed ammunition depots and barracks used by the Russians.
The latest update from the Ministry of Defence reads: “Russian artillery continues to strike the Sloviansk area of the Donbas from around Izium to the north and near Lysychansk to the east. Russian forces have likely made some further small territorial advances around Popasna.
"Fires from Izium continue to focus along the axis of the E40 main road. Control of the E40, which links Donetsk to Kharkiv, is likely to be an important objective for Russia as it attempts to advance through Donetsk Oblast.”