Russian military vehicles drive to the Kharkiv direction on a mission in Ukraine. File/AP
A Moscow-backed separatist leader in the east meanwhile said Russian forces were fighting "difficult" battles against Kyiv's troops in several parts of eastern Donetsk region.
A Ukrainian official also said Kyiv's troops were closing in on the eastern city of Lysychansk, captured by Russian troops after fierce artillery battles in July.
Moscow's announcement late Saturday of the pullback alongside Kyiv's claim to have entered the town of Kupiansk are the most significant shifts in battlefield dynamics after months of fighting in eastern Ukraine that has been dominated by Moscow.
"A decision was made to regroup Russian troops stationed in the Balakliya and Izyum regions to bolster efforts along the Donetsk front," Russia's defence ministry said in a statement.
The Russian military's Uragan multiple rocket launchers fire rockets at Ukrainian troops. File/AP
News of the drawdown came just after Ukrainian special forces published images on social media showing camouflage-clad officers with automatic weapons "in Kupiansk", a town of about 27,000 people.
Ukrainian troops had also liberated Vasylenkovo and Artemivka in Kharkiv region, President Volodymyr Zelensky said in his evening address on Saturday.
"These last days, the Russian army has shown us its best (side) — its back," he said. "There is no place in Ukraine for the occupiers. There never will be," he added.
Separately speaking to the annual Yalta European Strategy forum, Zelensky said Russia "is doing everything to break the resistance of Ukraine, Europe and the world during the 90 days of this (coming) winter", counting on an eventual weakening of Western support for Kyiv due to rising energy prices and heating problems.
"It's their final argument," he said.
Observers expect Ukrainian forces to make further gains in the Kharkiv region, which borders Russia and has been either controlled by Moscow-backed authorities or shelled by its artillery for months.
There was no official confirmation that Kyiv's troops had also routed Russian forces from Izyum — an important staging ground for Moscow's war effort with a population of around 45,000 people before the invasion.
A Russian Su-25 ground attack jet fires rockets on a mission at an undisclosed location in Ukraine. File/AP
But images flooding social media appeared to show Ukrainian forces within the city and Russian observers of the conflict said there were initial reports Moscow's army had already withdrawn.
"Ukrainian troops are advancing in eastern Ukraine, liberating more cities and villages. Their courage coupled with Western military support brings astonishing results," said foreign ministry spokesman Oleg Nikolenko.
"It's crucial to keep sending arms to Ukraine. Defeating Russia on the battlefield means winning peace in Ukraine," he added.
Ukraine's push appears to have caught Russian troops largely off guard.
State media on Friday published footage of Russian tanks, artillery and support vehicles moving towards Kharkiv in columns on dirt roads -- a bid to dispatch reinforcements to the region.
But a day later, Russia announced a three-day operation to redeploy forces, moving away from Kharkiv to the industrial Donbas region further south.
The capture of urban hubs like Kupiansk and Izyum would be a serious blow to Moscow's ability to resupply its positions on the eastern frontline, and could see Russia's hold over the east severely diminished.
Ukrainian soldiers attend their positions, in the Donetsk region, Ukraine. File/AP
In one village captured by the advancing Ukrainians, electric pylons were toppled, cables lay across the ground and houses were gutted, AFP journalists reported.
On the road towards the recaptured town of Balakliya, the journalists saw abandoned Russian armour painted with the letter "Z" -- a symbol of Russia's invasion.
Ukrainian troops were also advancing along portions of the southern front line, a spokesperson said Saturday, in some regions by dozens of kilometres, into territory captured by Russian troops at the beginning of the invasion.
Russian news agencies reported six large explosions in Nova Kakhovka, a town held by Russian troops in the southern Kherson region.
In the eastern Donetsk region, rebel leader Denis Pushilin said the situation in the town of Lyman was "very difficult" and that there was also fighting in "a number of other localities", particularly in the northern part of the region.
Russian president orders the Russian defence minister and the chief of the military’s General Staff to put the nuclear deterrent forces in a "special regime of combat duty.”
Governor Dmytro Zhyvytskyy posted photographs of the charred shell of a four-story building and rescuers searching rubble. In a later Facebook post, he said many Russian soldiers and some local residents also were killed during the fighting on Sunday.
Kathy Lueders, who heads the agency's human spaceflight program, told reporters on a call that operations on the research platform were proceeding "nominally" and "we're not getting any indications at a working level that our counterparts are not committed."
Muhammad Saeed Al Mulla contributed to the establishment of a number of non-oil Emirati institutions that have become the pillars of the diversified national economy.
The Ministry of Education makes the use of the Emirates Standardised Test (EmSAT) optional for universities starting from the admission procedures for the academic year 2023-2024.
Muhammad Saeed Al Mulla was born in Al Shindagha, Dubai, in 1926, and grew up in the emerging city, which was groping for its path as a center for trade and pearl hunting.