The eastern Lugansk region is almost entirely under Russian control and Moscow has claimed its full annexation.
The main city, also called Lugansk, came under a "massive" missile attack, according to its Russia-appointed head, Leonid Pasechnik.
The emergency situations ministry said a section of an apartment block had collapsed.
"Unfortunately three people were killed," the ministry said while rescuers managed to pull seven survivors from the rubble.
It posted images of a five-storey block of flats with the facade ripped open from the roof to the ground and a deep crater in the ground.
The regional government said on Telegram that 35 people were injured including three children aged 8 to 16, citing the regional health ministry.
Of these, "five people are in a serious state," said regional health minister Nataliya Pashchenko.
Pasechnik said on Telegram that "Ukrainian nationalists in the daytime launched a massive attack on civilian infrastructure of Lugansk."
"In one multi-storey block, a section has entirely collapsed. There are people under the ruins. Rescuers are working to free them," Pasechnik said.
Rescuers carried out one elderly casualty on a stretcher, footage posted by the emergency ministry showed.
The Russian defence ministry said Ukraine "deliberately fired five US-made ATACMS missiles at residential districts of the city of Lugansk".
"Four American missiles were shot down by Russian air defences. One missile struck two blocks of flats," the ministry said.
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