Deputy Defence Minister Hanna Malyar said Ukrainian forces advanced two kilometres (1.2 miles) in a week.
The claims signal a momentum shift in Bakhmut – but more widely, there is no clear evidence of the much-anticipated Ukrainian counter-offensive.
Russia has denied reports of Ukrainian gains, after claims from both sides.
“The individual declarations on Telegram about a ‘breakthrough’ on several points on the frontline do not correspond to reality,” Russia’s defence ministry said.
The general situation in the special military operation zone is under control,” it added, using Russia’s term for the invasion.
However, in a updated statement on Friday morning the ministry claimed Russian troops had changed their position for strategic reasons.
It said units of the southern group of Russian forces had taken up a better defensive position in the Maloilinivka area, something which took into consideration “the favourable conditions of the Berkhivka reservoir”.
Maloilinivka and Berkhivka are both near Bakhmut.
Responding to the statement, the head of Russia’s Wagner mercenary group Yevgeny Prigozhin said what the Ministry of Defence was talking about “is unfortunately called ‘fleeing’ and not a ‘regrouping'”.
As the intense, bloody battle has worn on, Bakhmut has become symbolically important – though many experts question its tactical value.
In a post on Telegram, Ms Malyar said Bakhmut has become a target of “almost sacred” importance to Russia.
She accused the Kremlin of simultaneously claiming false victories on one hand, while also spreading lies about being short on weapons and ammunition.
Describing the “real situation” over the past week, Ms Malyar claimed Russia suffered significant troop losses, as Ukraine gained 2km without losing any positions.
Earlier Prigozhin accused regular Russian troops of abandoning positions in Bakhmut.
“The situation on the flanks is developing according to the worst predicted scenario,” he said.
And Russian military bloggers reported Ukrainian advances or troop movements in several areas.
The institute for the study of war also said Ukrainian forces had probably made gains of 2km in Bakhmut.
And the US-based think tank said the Russian defence ministry acknowledging “the Ukrainian counterattacks uncharacteristically quickly” – by denying them – indicated “increased panic”.