There was no immediate independent confirmation that the fighting had ended, but the governor of Russia's Belgorod region said the anti-terrorism measures he introduced after Monday's attack had been lifted.
And one of two groups claiming to be behind the raid said in a post on social media: "One day we will return to stay."
The Russian military said it had killed more than 70 Ukrainian nationalists and destroyed four armoured vehicles. Belgorod Governor Vyacheslav Gladkov said one civilian had been killed "at the hands of the Ukrainian armed forces."
The fighting forced Russia to evacuate towns along the border with northeast Ukraine.
Russia has blamed Ukraine for the attack, which Kyiv has denied.
The two groups who claimed responsibility for the raid, the Freedom of Russia Legion and the Russian Volunteer Corps (RVC), both say they comprise armed Russian fighters seeking to overthrow President Vladimir Putin.
The Legion says it is recognized by Ukraine and its members have fought there against Russian forces. The RVC has claimed responsibility for previous attacks inside Russia, including a cross-border raid in the neighbouring Bryansk region in March.
Mark Galeotti, head of the London-based Mayak Intelligence consultancy, said the militant groups covered a variety of political perspectives united by wanting to see Putin's downfall, adding that the incursion was not a decisive military action but "a shaping operation, preparing the battlefield for the inevitable and probably looming Ukrainian counter-offensive."
Kyiv publicly denied blame for the raid, though some of its denials were pitched with apparent irony, to mimic past Russian denials of a role in separatist movements in Ukraine.
Moscow, which portrays its Feb. 24, 2022, invasion of Ukraine as a response to a security threat from Kyiv, said the attackers were Ukrainian saboteurs, seeking to deflect attention from Russia's capture of the city of Bakhmut three days ago, after the bloodiest land battle in Europe since World War Two.
US casts doubt on reports that American-supplied weapons were used in the attack in Russia
The United States is sceptical of reports that US-supplied weapons were used in an attack inside Russian territory, a State Department spokesman said on Tuesday, adding that Washington does not encourage or enable strikes inside Russia.
"Let me say a few things about it. Number one, we've seen some of the reports circulating on social media and elsewhere making claims that US-supplied weapons were used in these attacks. I will say that we're sceptical at this time about the veracity of these reports. As a more general principle, as we've said, and I believe I said yesterday, we do not encourage or enable strikes inside of Russia, and we've made that clear. But as we've also said, it is up to Ukraine to decide how to conduct this war," US State Department Spokesperson, Matthew Miller said.
Group claiming Belgorod raid says cross-border incursions may stretch Russian defences
A two-day incursion from Ukraine into Russia's western borderlands could push the Kremlin to divert troops from the frontline as Kyiv prepares a major counteroffensive, and deal Moscow a psychological blow, said a representative of one of the two groups claiming to be behind the Belgorod operation.
"Russia will be forced to bring reinforcements from other parts of the frontline. If they move any units from the Ukrainian frontline it will make things easier for Ukraine to advance in the counteroffensive and liberate its territory," Alexei Baranovsky, a spokesperson for the political wing of the Freedom of Russia Legion, told Reuters in Kyiv on Tuesday.
Though Kyiv has denied any role, the biggest cross-border raid from Ukraine since Russia invaded 15 months ago was almost certainly coordinated with Ukraine's armed forces as it prepares to attempt to recapture territory, experts said.
Ukraine says it plans to conduct a major counteroffensive to seize back occupied territory and Russia has built sprawling fortifications in Ukraine's east and south in readiness.
The incursion took place far from the epicentre of fighting in the eastern Donbas region and around a hundred miles from the frontlines in the northern Kharkiv region.
"These are the first steps in the main objective of overthrowing Putin's regime through armed force. There are no other alternatives," Baranovsky said.
Baranovsky said that he could not disclose the number of troops involved in the operation, but that the legion had four battalions in total.
Baranovsky denied there had been heavy losses and called Russian reports of large casualties disinformation.
He said the unit was part of Ukraine's International Legion and therefore part of its armed forces, but denied the incursion was coordinated with Ukraine.
Ukrainian presidential aide Mykhailo Podolyak repeated Kyiv's position that Ukraine had nothing to do with the operation.
Several similar incursions into Russia have taken place in recent months, but although this week's was the largest known so far, it is still tiny when compared to frontline battles.
Russia's Medvedev says Belgorod insurgents should be exterminated 'like rats'
Former president Dmitry Medvedev, now deputy chairman of Russia's Security Council, said Kyiv's denial was "lies" and that the attackers deserved to be exterminated "like rats."
Medvedev was speaking during an official visit to Laos where he held talks with senior government figures including President Thongloun Sisoulith.
"Responsibility ... is carried by the Kyiv regime, and ultimately by its sponsors overseas - that is, Washington and the countries of the European Union, along with affiliated states like Britain and others," Medvedev said, according to TASS.
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.
Reuters