Russian Defence Minister Sergei Shoigu has promised a “harsh response” to cross-border incursions from Ukraine.
His comments came after Moscow said it had defeated an attack in the Belgorod region.
However, regional governor Vyacheslav Gladkov said there had been a “large number” of drone attacks overnight.
Ukraine denies involvement in the raid – and two Russian paramilitary groups opposed to Russian President Vladimir Putin say they were behind it.
Attackers entered Russian territory from Ukraine on Monday.
Reporting to defence ministry officials on the incursion, Mr Shoigu said “more than 70 Ukrainian nationalists” had been killed and the rest pushed back into Ukraine.
“We will continue to respond to such actions by Ukrainian militants promptly and extremely harshly,” he said.
The two Russian paramilitary groups – the Russian Volunteer Corps (RDK) and Liberty of Russia Legion (LSR) denied that they had sustained any casualties, and said a Russian motorised rifle company had been destroyed.
The casualty claims by the warring sides have not been independently verified.
Russia also says that Western military vehicles were used in the incursion.
It posted pictures of destroyed US vehicles apparently at the scene of the fighting but some Ukrainian military experts and bloggers have suggested they could have been staged.
The US said it was sceptical that reports of US-supplied weapons being used in the incursion were true and did not “encourage or enable strikes inside of Russia”.
But Kremlin spokesman Dmitry Peskov said the vehicles were evidence of growing Western military involvement in Ukraine.
“It is no secret for us that more and more equipment is being delivered to Ukraine’s armed forces. It is no secret that this equipment is being used against our own military,” he said.
“We are drawing the appropriate conclusions.”
Meanwhile attacks in the region appear to be continuing.
Mr Gladkov said overnight attacks by drones were mostly dealt with by air defences, but some damage was caused to cars, private houses and administrative buildings in and around Belgorod city, as well as in the border district of Borisovka.
No-one was injured in the attacks, he added.
A “small fire” began after a gas pipeline was damaged in Grayvoron district.
Low-level attacks in Russian border regions such as Belgorod and Bryansk have become frequent in recent weeks.
Mr Gladkov said that agricultural workers were going out into the fields wearing helmets and bulletproof vests because of the threat of attacks.
Villages in Belgorod near the border were evacuated on Monday after coming under fire.
Monday’s raid led Moscow to declare a counter-terrorism operation, giving the authorities special powers to clamp down on communications and people’s movements.
The measures were only lifted the following afternoon, and even then, one of the paramilitary groups was claiming it still controlled a small piece of Russian territory.
