A BBC team on the ground confirmed the top floor of a large residential tower block was almost completely destroyed in the strike on Friday evening.
A building belonging to Ukraine's security services (SBU) was also hit, according president Volodymyr Zelensky.
He blamed "Russian missile terror" for the damage to the buildings.
Mr Zelensky posted a video on Telegram showing smoke rising from damaged buildings and a fire at street level.
He said he had convened emergency meetings with the SBU, interior ministry, emergency services and local officials following the incident.
Regional Governor Serhiy Lysak said that two children, aged 14 and 17, were among the injured. He added that they were being treated at home.
No fatalities have been reported following the strike, which happened at 20:30 (17:30 GMT) according to Mr Lysak.
The latest strike marked the third time the SBU building had been targeted by Russia, according to Dnipro Mayor Borys Filatov.
Both buildings were largely empty, he said, adding that the residential building had recently been completed and units were being put up for sale.
War in maps: Fighting in south Ukraine intensifies
It comes after Russia said on Friday it had intercepted two Ukrainian missiles over its southern Rostov region, bordering Ukraine.
Moscow said 15 people were hurt by debris falling in the southern port city of Taganrog.
The Russian defence ministry said the first S-200 missile was aimed at "residential infrastructure" in the city of around 250,000 people.
Shortly after, it said it downed a second S-200 missile near the city of Azov, with debris falling in an unpopulated area.
Rostov region Governor Vasily Golubev said 15 people suffered "light injuries" from shards in an explosion near the "Chekhov Garden" cafe in central Taganrog.
Taganrog is located on the coast of the Sea of Azov and about 25 miles (40km) from the border with Ukraine.
The city is also on a road leading to the port city of Mariupol, the strategic port city devastated by Russian shelling.
By Marita Moloney