The General Staff of the Ukrainian Armed Forces said the Russians had tried to close in on the city with a pincer movement, but that the fresh attempt to encircle Bakhmut from the north and south failed.
"The enemy continued his efforts to capture the city, with significant losses of troops and weapons," wrote the General Staff in Kyiv on Facebook in its daily situation report.
The military also said there had been an overall "decrease in the momentum" of attacks on Bakhmut by the Russian forces.
The battlefield claims could not be independently verified.
The British Defence Ministry had earlier said that there was "a chance" that the Russian assault on Bakhmut was "losing the limited momentum it had obtained." This is in part due to the re-deployment of some Russian units to elsewhere on the front, the analysts said.
Earlier, Ukrainian President Volodymyr Zelensky visited the front line near Bakhmut, where he presented medals to soldiers and went to a military hospital with the governor of the Donetsk region, Pavlo Kyrylenko.
Bakhmut has been the site of intense fighting between Ukrainian and Russian forces for months.
If Moscow's forces manage to overpower the town, analysts say this would open a route to the major cities of Slovyansk and Kramatorsk for Russian troops. That would bring them closer to the complete conquest of the Donetsk region, one of the Kremlin's stated aims since starting the full-scale war.
Zelensky later travelled to Kharkiv, where he praised the city for its resistance to sustained attacks by Russian forces last year.
He presented Mayor Ihor Terekhov with the insignia of a "Hero City of Ukraine," in honour of residents' resistance to Russian attacks last year.
"Kharkiv is a real hero city," Zelensky said, according to the UNIAN news agency. "Thanks to its citizens, this beautiful city, along with other cities, is defending our independence."
A Ukrainian counteroffensive in Kharkiv in May last year prevented Russian troops from pushing further into the country's north-east. The fighting led to high losses on both sides.
Following his visit, Zelensky spoke of the destruction but also the hope that he had seen.
"It is painful to see the towns in the Donbass over which Russia has brought terrible suffering and ruins," he said in his evening video address, mentioning "hourly air raid sirens, constant threat of shelling, a constant threat to life."
But despite the severe destruction and suffering, there is hope in those areas. "You can feel it," he said.
"We will do everything so that the blue and yellow colours can continue their liberation movement and normal life can return to our whole country, from Donetsk to the border," he added, referring to the nation's flag.
He also announced Kyiv's response to Russia's recent attacks targetting cities with combat drones. "We will definitely respond to any attack by the occupiers on our cities," Zelensky said.
"To all Russian attacks we will respond militarily, politically and legally."
At least 14 people were killed in Russian attacks including along the front line in the east and south of Ukraine, and 24 were injured, according to the Ukrainian army press service.
Zelensky referred to the attacks as a new "Russian terror."
"I saw a lot of destruction today in the surrounding area. But the most important thing is victory," he said in an earlier video. Everything will be rebuilt, he promised.
Meanwhile, Russia's anti-aircraft defence and sailors from its Black Sea fleet shot down three Ukrainian drones over Crimea, a Russian official on the Russian-occupied peninsula said.
The shootings caused explosions which broke windows, but there were no casualties, according to Mikhail Razvozhayev, who is governor of Sevastopol in the south-western part of Crimea.
"Warships were not hit," he said.
Russia took control of the peninsula in 2014 and launched an all-out invasion of Ukraine some 13 months ago. Ukraine is determined to free Crimea from Russian control, along with other territory more recently occupied.
Incidents involving drones have occurred in various parts of Crimea, sometimes causing serious damage, injuries and even deaths. The drone attacks have forced Russia to significantly increase its military effort in Crimea.
