Interior Minister Arseny Avakov said the security forces were successfully bringing separatist strongholds in the city "under control".
Five pro-Russian rebels were reported killed and at least four government soldiers injured in the city.
Rebels elsewhere in the region have confirmed they now have three tanks.
The appearance of the tanks, filmed in various towns in the Donetsk region, sparked a row between Kiev and Moscow, with Ukrainian President Petro Poroshenko protesting to his Russian counterpart, Vladimir Putin.
Russia denied Ukrainian allegations that the tanks had entered Ukraine from its territory.
Confirming that four soldiers were injured, Ukraine's National Guard said on its website that its recently formed Dnepr and Azov "volunteer" units were being used in the operation.
A rebel source told Russia's Interfax news agency that five fighters had been killed and that fighting was still under way.
The pro-Russian separatists in the Donetsk and neighbouring Luhansk regions declared independence after holding referendums last month which were deemed illegal by the government in Kiev. Ukraine and the West accuse Russia of supporting and arming the rebels - a claim denied by Moscow.
The rebellion began after February's ousting of the elected President, Viktor Yanukovych, whose last-minute decision not to sign a landmark treaty with the EU in November sparked mass street protests in Kiev.
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