Kyiv also ordered children to be evacuated from five settlements near the focus of their assault, as Russia claimed to have repelled Ukrainian forces.
Kyiv launched a grinding counteroffensive in June after stockpiling Western-supplied weapons and building up assault battalions, but progress has been slow.
Military spokesman Andriy Kovalyov said Ukrainian forces were edging further in the Zaporizhzhia region, which Moscow claims is part of Russia.
"Ukrainian forces had successes in the direction of Novodanylivka to Verbove," he told state media on Tuesday, naming two hamlets in the war-battered region.
He added that the troops were holding captured territory and attacking Russian artillery.
Ukrainian troops have also been trying to surround the eastern town of Bakhmut, which was captured by Russian forces in May.
The Russian-installed head of the Donetsk region, where Bakhmut is located, on Tuesday played down the Ukrainian push, after Kyiv claimed successes.
"The flanks are being held. The situation there is already stabilising," Denis Pushilin told Russian state media.
In the nearby Russian-controlled town of Gorlivka, its Moscow-aligned mayor said Ukrainian shelling had killed three civilians.
Mayor Ivan Prikhodko described the fatal shelling on a milk production facility as "horrifying".
Compared to Ukrainian offensives last year in the Kherson and Kharkiv regions, this time Kyiv's forces are crashing into Russian defensive lines of trenches and minefields that are kilometres deep.
But analysts say the capture of Robotyne is evidence that Ukrainian forces can puncture Russian lines as they push south.
The Ukrainian ministry responsible for reintegrating occupied territories announced an evacuation order in five settlements in the Zaporizhzhia region, citing the uptick in fighting.
Some 54 children and nearly 70 people accompanying them will be removed from the villages of Guliaipole, Stepnogirsk, Preobrazhenka, Yegorivka, and Novopavlivka, it said.
The ministry also said that the remains of 84 killed Ukrainian soldiers had been recovered from Russia.
"After identification, the bodies of our defenders will be handed over to their relatives for a dignified burial," it said in a statement.
President Volodymyr Zelensky meanwhile paid tribute to fallen soldiers on Tuesday to commemorate Ukraine's remembrance day.
"Words are never enough to express what we feel," he said in a statement, describing the those who were killed as people "whose lives became the life of Ukraine".
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