STORY: Mayor Alexey Kulemzin said on the Telegram messaging platform that the four died in a central part of the city.
Fragments of French-made Caesar howitzer shells were found, Kulemzin added.
Yulia, a resident of Donetsk, said there were 10 loud explosions.
On the same day, soldiers launched mortars at Ukrainian forces in the Avdiivka area.
A masked soldier with the call-sign "Moscow" said he was sent to the area as a conscript soldier.
"Their artillery is strong, I 'm not arguing," he added.
In recent weeks, Ukrainian forces have retaken territory previously held by Russian forces in the northeastern Kharkiv region.
East and south of there, Russia has concentrated forces for months in the Luhansk and Donetsk provinces, to expand territory held by Moscow-backed separatists since 2014.
This content was produced in a region of Ukraine backed by Russia, whose law restricts coverage of Russian military operations in Ukraine.