Ukrainian forces outnumbered Russians by eight to one in last week's counter-attack in the Kharkiv region, Russia's top occupation official there said, as per media reports.
Vitaly Ganchev told Russian TV that Ukraine's army had taken villages in the north and broken through to the Russian border, BBC reported.
Ukraine says it has regained control over 3,000 sq km of territory in a potential breakthrough in the war.
The Ukrainian army says it took back 20 villages in the past 24 hours alone, in its continued counter-offensive in the north-east of the country, BBC reported.
It also said its forces have taken control of around 500 sq km in the southern Kherson region of the country.
Also read | Ukraine successes have 'significant implications' for Russian operation
UK defence officials say the Ukrainian army's successes will have "significant implications" for Russia's overall operational design.
Kremlin spokesman Dmitry Peskov was, however, undeterred, saying operations in Ukraine would continue "until all the tasks that were initially set" had been fulfilled.
President Vladimir Putin is constantly being updated with the latest developments, he added.
Russia said its forces were carrying out strikes in those areas that Ukraine had retaken recently.
This included targets in Izyum and Kupiansk which were taken by Ukraine on Saturday. Russia confirmed its forces' retreat from both towns, which it said would allow them to "regroup".
Valerii Marchenko, mayor of Izyum, told the BBC the Ukrainian army was in his city and the state flag had been raised.
The military is now engaged in cleaning up the war-torn city and Ukrainian forces are searching for Russian soldiers potentially hiding in people's houses.
Also read | Ukraine says it has tripled retaken area in rapid counteroffensive
Marchenko said that after "about 10 days", residents who had to flee the city would be able to return "safely", BBC reported.
Russia has been accused of targeting civilian infrastructure in revenge for setbacks on the battlefield.
A wave of missile strikes on Sunday caused power cuts across the region.
