The comments by Oleksiy Danilov, secretary of Ukraine's National Security and Defence Council, were Kyiv's latest positive assessment of the month-old counterattack although Moscow has not acknowledged Ukraine's gains.
Ukraine's Deputy Defence Minister Hanna Maliar also reported gains around the shattered city of Bakhmut despite fierce Russian resistance.
Russian forces had captured it in May after 10 months of battles.
Russia, which began its full-scale invasion in February 2022, still holds swathes of territory in eastern and southern Ukraine but Ukrainian President Volodymyr Zelenskiy said on Monday his troops had made progress after a "difficult" week.
"At this stage of active hostilities, Ukraine's defence forces are fulfilling the number one task - the maximum destruction of manpower, equipment, fuel depots, military vehicles, command posts, artillery and air defence forces of the Russian army," Danilov, head of Ukraine's National Security and Defence Council, wrote on Twitter.
"The last few days have been particularly fruitful," he said, without providing details.
Southern military command spokesman Valeriy Shershen said Ukrainian troops had advanced by up to two kilometres in the Berdiansk direction despite fierce Russian resistance.
Deputy Defence Minister Maliar said Ukrainian forces were making gains every day in areas outside Bakhmut.
"We are advancing on the southern flank of Bakhmut," Maliar told national television.
"To the north, to be honest, there is heavy fighting and so far no advance.
"... Our forces are encountering serious resistance.
"The enemy is pouring in all its forces to stop in the south and in the east."
She said Russian forces were also making advances further north, near Lyman and in Svatove.
In Makiivka, a town in the Russian-controlled part of the Donetsk region, Ukraine's military said it had destroyed a formation of Russian forces.
Russia-installed officials said one civilian died and 36 were injured in the attack.
Reuters has been unable to verify the situation on the battlefield.
Each side says the other is suffering heavy losses.
Accounts of frontline fighting from the Russian Defence Ministry said its forces had foiled Ukraine's in five areas of the eastern Donetsk region.
It also reported repelling attacks near Lyman and disrupting enemy operations in the Zaporizhzhia region, where Ukraine says its forces have captured a cluster of villages.
The General Staff of Ukraine's military reported success in repelling Russian attacks in Kupiansk in the north, Bakhmut and near the contested towns of Avdiivka and Maryinka to the south.
Russia said on Tuesday Ukraine had attacked Moscow with at least five drones that were all shot down or jammed, although one of the capital's airports re-routed flights for several hours.
The two countries also accused each other of plotting to stage an attack on the Russian-held Zaporizhzhia nuclear power station.
Zelenskiy said he told his French counterpart, Emmanuel Macron, about Russian "dangerous provocations" at the plant in southeastern Ukraine.
He said he and Macron had "agreed to keep the situation under maximum control together with the IAEA" - the United Nations International Atomic Energy Agency.
Zelenskiy and the Ukrainian military provided no evidence for their assertions.
Russian troops seized the station, Europe's largest nuclear facility, in the days following the Kremlin's invasion of Ukraine in February 2022.
Renat Karchaa, an adviser to the head of Rosenergoatom, which operates Russia's nuclear network, said Ukraine planned to drop ammunition laced with nuclear waste transported from another of the country's five nuclear stations on the plant.
He offered no evidence in support of his allegation.
