Vitaly Ganchev told Russian TV that Ukraine's army had taken villages in the north and broken through to the Russian border.
Ukraine says it has regained control over 3,000 sq km (1,158 sq miles) of territory in a potential breakthrough in the six-month war.
The BBC cannot verify these figures.
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.
It also said its forces have taken control of around 500 sq km in the southern Kherson region of the country.
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.
Mr Marchenko said that after "about 10 days", residents who had to flee the city would be able to return "safely".
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.
The blackouts reportedly affected millions of people in the Kharkiv and Donetsk regions.
Ukraine's Defence Minister, Oleksii Reznikov, said the priority now was to secure the territorial gains made in a week of rapid advances in the Kharkiv region.
The Russian army appears to have abandoned large quantities of equipment and ammunition as it withdrew from areas it had held since the first weeks of the war.
A Russian battle-tank maker has switched to "round-the-clock" production after these setbacks, according to reports.
Workers at Uralvagonzavod in Nizhny Tagil, Russia's largest armoured vehicle manufacturer, have reportedly not been allowed to go on vacation due to the "production necessity".
Ukraine's rapid progress seems to have stunned Russia. The Kremlin says its troops are regrouping, but pictures from some liberated areas in the Kharkiv region suggest a hasty departure by the invading forces, with military vehicles, ammunition, and equipment left behind.
If the gains are confirmed - and if they hold - it's the most significant change on the frontlines since Russian troops left the Kyiv region five months ago. The extraordinary advance would mark a humiliating setback for Russia, and be a boost to Ukraine's position that it can push the Russians out while asking for more Western weapons.
President Zelensky has talked about a possible breakthrough, suggesting that more advances could happen before winter. But the country still faces huge challenges. Around a fifth of it remains under occupation and, in the south, Ukrainian troops have reportedly faced more resistance in their offensive, as Russia has fortified its positions.
Ukraine feels it has the momentum, and is pushing forward. But questions remain about how far it can go.
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