According to Ukrainian sources, on Saturday, troops entered Kupiansk, a crucial supply base for Russian forces in the east.
The Russian defence ministry then declared that its troops had left nearby Izyum so they could "regroup."
The ministry has acknowledged the removal of troops from Balaklyia, a third significant town, in order to "bolster efforts" on the Donetsk front.
If maintained, the Ukrainian advances would be the biggest since Russia retreated from regions near Kiev in April.
President Volodymyr Zelensky asserted in his nightly video message on Saturday that since starting a new counteroffensive earlier this month, Ukraine had liberated 2,000 sq km (700 sq miles) from Russia.
Given that it is twice the area of land Mr Zelensky claimed had been liberated when he spoke on Thursday evening, his claim would imply that half of that area has been reclaimed in the last 48 hours alone.
Because Izyum served as a significant military hub for Moscow, the admission by Russia that it has left the area is significant, according to a BBC report.
"A three-day operation was carried out on the drawdown and organised transfer of the Izyum-Balakliya group of troops to the territory of the Donetsk People's Republic," the Russian statement said.
"In order to prevent damage to the Russian troops, a powerful fire defeat was inflicted on the enemy."
According to the Russian state-run Tass news agency, the head administrator of the Kharkiv region's areas under Russian authority soon suggested that its residents leave for Russia in order to "save lives."
Additionally, the governor of the nearby Russian region of Belgorod declared that those waiting in line to cross the border would have access to mobile food, heating, and medical care.
The gains will be interpreted as evidence that Ukraine's army is capable of retaking Russian-occupied territory, which is critical as Kyiv continues to request military assistance from its hard-pressed Western partners.
Dmytro Kuleba, the foreign minister of Ukraine, claimed that recent events had demonstrated that his country's forces were capable of defeating the Russian army and that more Western armaments would hasten the end of the war.