The tanks entered Ukraine along with other armour through a checkpoint controlled by rebels in the Luhansk region, Arsen Avakov said.
Ukraine’s army engaged two of them and destroyed part of the column, he said.
Russia has rejected Arsen Avakov’s statement as “another fake piece of information”.
Meanwhile, Ukrainian President Petro Poroshenko, who was elected last month, and his Russian counterpart Vladimir Putin had a “substantial” conversation by phone on Kiev’s plan of peaceful settlement in the east, Ukraine’s [residency said in a statement.
Petro Poroshenko said that it was “unacceptable” for the tanks to be crossing into Ukraine, his press secretary said.
Pro-Russian separatists in the Donetsk and 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.
Arsen Avakov said the tanks had crossed the border from Russia along with armoured personnel carriers and artillery pieces in the Dyakove area of Luhansk region, before moving into the neighbouring Donetsk region.
There, Ukraine’s interior minister said, the tanks headed for the town of Snizhne on Thursday morning. Two then proceeded to the town of Horlivka and were attacked by government forces.
“The fight is under way,” Arsen Avakov said.
“I cannot say about its final outcome, but part of this column has been destroyed.”
Unverified video has been posted on YouTube of a battle tank rolling down a street said to be in Snizhne. The footage was shot from a flat overlooking the street.
A second clip, also said to have been recorded in Snizhne, shows a similar tank passing a roundabout.
A rebel spokesman told Russia’s Ria-Novosti news agency that they were fighting a force of 40 Ukrainian tanks north of the city of Luhansk.
The tanks had been stopped at the village of Makarovo, the spokesman said.
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