Border guards halted the advance of the column about 5 km northeast of the town, which is on the Azov Sea, it said in a Facebook post.
The new military thrust by the rebels, which a pro-government militia commander said might be aimed at capturing the government-held port city of Mariupul, appeared to open up a new southern front in the five-month conflict in Ukraine's east.
Fighting has been hithero concentrated around the two big rebel-held cities of Donetsk and Luhansk to the north, Reuters reported.
The commander of a Ukrainian national guard unit in the area near Novoazovsk where the fighting was reported to be happening told Reuters by telephone: "A war has broken out here." He said he could not speak and ended the conversation.
Semen Semenchenko, commander of the pro-government Azov militia, said on his Facebook page that around 50 armored vehicles had crossed the border from Russia.
About 40 of them were trying to move in the direction of Mariupol while the remainder were moving north towards Amvrosiyivka, he said.
"The invasion of the Russian occupiers is taking place," he said. The attacking forces were, for now, "localized and easily neutralized" and there had been up to now no fighting around Mariupol itself, he added.
The column's movement towards Novoazovsk had been accompanied by artillery shelling from across the border, he said.