The Ukrainian authorities did not say whether this was the start of a long awaited counter-offensive.
Meanwhile, Russia's military said it repelled Monday's attack.
Bakhmut has for months been at the heart of fierce fighting. It has little strategic value - but is important symbolically both for Kyiv and Moscow.
Monday's claims by Ukraine and Russia have not been independently verified.
Earlier in the day, Ukrainian military sources told the BBC that a series of small-scale armoured offensive operations were under way.
In a post on social media, Ms Maliar said that "despite stiff resistance and the enemy's attempts to hold the their positions, our military units advanced in several directions during the fighting".
She said in Orikhovo-Vasulivka and Paraskoviivka, Ukrainian troops gained from 200m to 1,600m (656-5,250ft), while in Ivanivske and Klishchiivka they advanced between 100m and 700m (330-2,300).
All four villages are located within several kilometres of Bakhmut. The battle for the city in the Donetsk region has been the longest and bloodiest of the war.
In a video address late on Monday, President Volodymyr Zelensky thanked Ukrainian fighters for delivering "the news we are expecting" on the Bakhmut direction.
The Russian paramilitary group Wagner had claimed to have captured the city in late May. In recent weeks, some analysts have suggested Kyiv's forces are attempting to encircle Bakhmut and trap Russian units.
A major Ukrainian counter-offensive has been long awaited but Kyiv has already said it would not give advance warning of its start.
There has been a notable increase in military activity, with Ukraine claiming to have made marginal gains elsewhere on the front line.
The latest reports are being seen as a fresh sign that the expected Ukrainian push may have begun.
The Russian defence ministry said on Monday that a new attack by Ukrainian forces in the Donetsk had been repelled.
It said the attacking side suffered heavy casualties, and that 28 tanks - including eight German-made Leopards - were destroyed.
This comes a day after Moscow said a Ukrainian "large-scale offensive" in the Donetsk region had begun on Sunday - but was unsuccessful. Ukraine's military said it had no information about such a major attack in the region.
Video of what Russia said was the battle in Donetsk showed military vehicles under heavy fire in fields. Russia claims it killed 300 troops and destroyed 16 tanks.
"We do not have such information and we do not comment on any kind of fake," a Ukrainian army spokesperson told Reuters.
There has been a significant increase in Ukrainian messaging on when and how their counter-offensive could take shape.
Ukraine has been planning such a move for months. But it has wanted as much time as possible to train troops and to receive military equipment from Western allies.
Foreign Minister Dmytro Kuleba said his country now had enough weapons for a counter-offensive but would not comment on whether it had begun, Reuters reported.
Officials in Kyiv have warned against public speculation over the offensive, saying it could help the enemy.
"Plans love silence. There will be no announcement of the start," the defence ministry said in a recently posted video. It featured masked and well-armed troops holding their fingers against their lips.
It will take Ukraine time to achieve its goal of liberating territory taken by Russia as far back as nine years ago.
Much is at stake because the government in Kyiv needs to show the people of Ukraine - and Western allies - that it can break through Russian lines, end the effective military deadlock and recapture some of its sovereign territory.