Fighting is reportedly ongoing in the area, as Moscow said troops supported by 11 tanks and more than 20 armoured combat vehicles, crossed the border near the town of Sudzha, 10km (six miles) from the frontline.
Thousands of people have left their homes in the region, officials said. Ukraine has yet to comment on the Russian allegations.
Speaking ahead of a meeting of the Security Council in Moscow, Mr Putin accused Ukrainian forces of "firing indiscriminately" at civilian buildings and residences.
Fighting reportedly took place in various villages on Russian territory throughout Tuesday. It was followed by Ukrainian air attacks which killed three civilians and continued into the night, Russian authorities said.
Twenty-four people, including six children, have been wounded in Ukrainian shelling of the border region, Moscow said.
On Wednesday, Russia’s Ministry of Defence claimed it prevented the Ukrainian Armed Forces from advancing “deep into Russian territory” in the Kursk region and said it had destroyed several Ukrainian drones overnight.
However, a number of air alerts continued to be issued in Kursk, where local authorities urged residents to limit their movements and all public events were cancelled.
Footage posted online - and verified by the BBC - showed fighter jets flying low overhead in the region on Tuesday, with smoke rising from areas on the ground.
The acting regional governor, Alexei Smirnov, said he had briefed Russian President Vladimir Putin about the situation, which he said was under control.
Mr Smirnov also said several thousand people had left areas of the region that were under attack and added doctors from Moscow and St Petersburg were on their way to offer assistance.
Kyiv has not yet commented on any of the reports about events in Kursk.
One colonel in Ukraine’s military, Vladislav Seleznyov, told the prominent Nexta channel the attack was “preventative” with an estimated 75,000 Russian troops continuing to gather close to the border.
After a major cross-border incursion by Russia into the north-eastern Kharkiv region in May, there had been fears Moscow would attempt the same into the Sumy region further north.
With Ukraine now apparently capturing several settlements and highways the other way, those ambitions may well have been frustrated, for now.
But with Ukrainian forces already overstretched and outmanned, some military analysts are questioning the wisdom of such cross-border raids.
This isn't the first incursion into Russia by fighters based in Ukraine. Some groups of anti-Kremlin Russians launched raids last year, which were repelled.
The forces crossed into the Belgorod and Kursk regions again in March, where they engaged in clashes with Russian security forces.
