One bomb directly hit a medical facility, while several others exploded on hospital grounds and near other health care buildings, Kramatorsk City Military Administration head Oleksandr Honcharenko said.
Despite the extensive damage, no patients were injured, he said.
Another bomb struck an apartment building. One person was initially reported killed and trapped beneath the rubble, while five civilians were reported injured.
One of the injured was in serious condition and evacuated to Dnipro. The others were receiving medical care, Honcharenko said.
Nine apartment buildings and three health care facilities were damaged in the attack. Authorities were still assessing the full extent of the destruction.
The death toll later rose to two, while the number of injured increased to 10, Donetsk Oblast Governor Vadym Filashkin said. Three medical facilities were damaged, he added.