It said 30 people died from carbon dioxide inhalation, while eight others died from the impact of jumping out of windows of the building of regional council of trade unions, Xinhua reported.
Up to 50 people, including 10 policemen, have sought medical assistance due to the blaze.
Kiev blamed on the Right Sector, a Ukrainian ultra-nationalist group, and self-defence militants for the incident. Police told Russia's Itar-Tass news agency that riots emerged earlier in the day between football fans from the eastern city of Kharkov and Right Sector radicals who had come from Kiev.
30 people died from carbon dioxide inhalation
The clashes came the same day as Ukraine initiated a military assault in its eastern city of Slaviansk in a bid to retake government buildings occupied by pro-Russia activists.
Ukraine's defense ministry has confirmed the loss of two helicopters. It said the aircraft were shot down by a surface-to- air missile, which proved the activists were not civilians but well-equipped and well-trained militants.
Kremlin said the assault meant Kiev had "crossed out" the Geneva agreements reached April 17.
Calling the military move a "manifestation of a crime of feebleness of Kiev authorities," Russian Prime Minister Dmitry Medvedev urged those in power to "come to their sense...otherwise the country's fate might be completely sad."