In what appeared to be one of the biggest incursions from Ukraine since the war began 15 months ago, two purported anti-Kremlin armed groups employing Russians based abroad said they were responsible for the attack in Russia's Belgorod region.
The Russian defence ministry, which blamed the Ukrainian authorities, said its forces had surrounded the enemy fighters and defeated them with "air strikes, artillery fire and active action by border units".
It said in a statement that more than 70 Ukrainian fighters had been killed, and four armoured vehicles and five pick-up trucks destroyed.
"The remnants of the nationalists were pushed back to Ukrainian territory, where they continued to be hit by gunfire until they were completely eliminated," the ministry added.
The Belgorod regional governor said one civilian had been killed "at the hands of the Ukrainian armed forces".
Reuters was unable to verify the assertions.
One of the two fighting groups - the Russian Volunteer Corps (RVC) - said on social media: "One day we'll come to stay."
Ukraine's government had said it was watching the situation but had "nothing to do with it".
It said the same in March when one of the groups - which Russia said consisted of ultranationalist Russian extremists managed by Ukrainian intelligence - mounted an incursion into another border region.
Former president Dmitry Medvedev, now deputy chairman of Russia's Security Council, said Ukraine's denial was "lies" and that the attackers deserved to be exterminated "like rats".
Belgorod governor Vyacheslav Gladkov said he had cancelled a "counter-terrorism operation" regime that had since Monday allowed authorities to introduce a host of restrictions on freedom of movement and communication.
Earlier on Tuesday, he had said the army and other security forces were still mopping up, and had urged residents who had been relocated not to return yet.
He said one elderly woman had died while being moved and three people were being treated for various injuries.
Later he added, without providing detail, that a civilian had been killed.
The two groups that claimed responsibility for the incursion are the RVC and the Freedom of Russia Legion - a Ukraine-based Russian militia led by Russian opposition figure Ilya Ponomaryov that says it is working inside Russia to overthrow President Vladimir Putin.
Ponomaryov is wanted in Russia, where the authorities have accused him of spreading false information about the army and designated him a terrorist.
The RVC, which claimed responsibility for the March incursion, was founded last August by Ukraine-based Russian nationalist Denis Kapustin, and announced on May 17 that it was joining forces with the Legion.
Kremlin spokesman Dmitry Peskov said Russia was conducting its "special military operation" in Ukraine in part to ensure that such incidents could not be repeated.
"This once again confirms that Ukrainian militants are continuing their activities against our country. This requires a great deal of effort from us, and these efforts are continuing."
Ukraine and its allies say Russia's military campaign is an unprovoked invasion to grab territory from Ukraine, independent since the Russian-dominated Soviet Union broke up in 1991.
Russia says it is acting to defend its own security from what it says is an ever-expanding NATO.
Asked about reports that the attackers were ethnic Russians rather than ethnic Ukrainians, Peskov said: "They are Ukrainian fighters from Ukraine. There are many ethnic Russians living in Ukraine. But they are still Ukrainian militants."
