The Russian Foreign Ministry told the BBC that the claim was "another fake piece of information."
According to Avakov, tanks crossed the border at a checkpoint controlled by pro-Russian separatists in the Luhansk region of eastern Ukraine.
Armored vehicles and artillery were part of the columns, Avakov said, citing Ukrainian intelligence.
Two of the tanks split off and headed in the direction of Gorlovka, also known as Horlivka, and were attacked by Ukrainian military forces, Avakov said.
Part of that column was destroyed, he said.
The countries have been engaged in a tense stand-off since March, when Crimea joined to Russia.
BA/BA