Moscow, which has the initiative on the battlefield, has repeatedly refused calls by Ukraine, Europe and US President Donald Trump for a full and unconditional ceasefire.
At May 2 talks in Istanbul it demanded Kyiv pull troops back from the frontline, agree to end all Western arms support and give up on its ambitions to join the NATO military alliance.
Dnipropetrovsk is not among the five Ukrainian regions over which Russia has asserted a formal territorial claim.
It is an important mining and industrial hub for Ukraine and deeper Russian advances into the region could have a serious knock-on effect for Kyiv's struggling military and economy.
Dnipropetrovosk had an estimated population of three million before Russia launched its offensive. Around one million people lived in the regional capital, Dnipro.
Russia's defence ministry said forces from a tank unit had "reached the western border of the Donetsk People's Republic and are continuing to develop an offensive in the Dnipropetrovsk region".
The advance of Russian forces into yet another region of Ukraine is both a symbolic and strategic blow to Kyiv's forces afer months of setbacks on the battlefield.
There was no immediate response from Ukraine to Russia's statement.
Moscow in 2022 claimed annexation of the frontline Donetsk, Kherson, Lugansk and Zaporizhzhia regions, which it did not have full control over.
Already in 2014, it seized the Crimean peninsula following a pro-EU revolution in Kyiv.
In a set of peace demands issued to Ukraine at the latest talks, Moscow demanded formal recognition that these regions were part of Russia -- something Kyiv has repeatedly ruled out.
- Strategic setback -
Tens of thousands have been killed in Russia's three-year offensive, millions forced to flee their homes and cities and villages across eastern Ukraine devastated by relentless air attacks and ground combat.
In more than a decade of conflict with Kremlin-backed separatists and the Russian army, Ukraine has never had to fight on the territory of the Dnipropetrovsk region until now.
Russia's ex-president Dmitry Medvedev, now deputy chairman of the national security council, said the fresh advance was a warning to Kyiv that it should give in to Russia's demands at peace talks.
"Those who do not want to recognise the realities of the war at negotiations, will receive new realities on the ground. Our armed forces have started an offensive in the Dnipropetrovsk region," he said on social media.
Russia's army posted photos showing troops raising the Russian flag over the village of Zorya in Ukraine's Donetsk region, close to the internal border.
Ukrainian military personnel previously told AFP that Russia could advance relatively quickly in the largely flat region, given there were fewer natural obstacles or villages that could be used as defensive positions by Kyiv's forces.
The region -- and in particular the city of Dnipro -- have been under persistent Russian strikes for the last three years.
Russia used Dnipro as a testing ground for its "experimental" Oreshnik missile in late 2024, claiming to have struck an aeronautics production facility.
Earlier on Sunday local Ukrainian officials said one person was killed in the region in an attack on a village close to the frontline.
Moscow also continued to accuse Ukraine of refusing to agree to take back the bodies of killed soldiers, after the two countries traded accusations a day earlier for thwarting a prisoner exchange agreed at talks in Istanbul.
Russia's defence ministry said trains carrying corpses were headed to the border point, where more than 1,200 had arrived on Saturday in refrigerated trucks.
Ukraine said on Saturday that the two sides had never agreed a date or time for some 6,000 bodies in total to be handed back.
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