Russia said it inflicted heavy casualties in return.
The clashes took place in Ukraine's Kherson region, where Russian forces rolled up major gains early in the war.
Ukrainian authorities kept the world guessing about their intentions, sidestepping talk of a major counteroffensive over the past couple of days.
The port city of Kherson, with a prewar population of about 300,000, is an important economic hub close to the Black Sea and the first major city to fall to Russian forces in the war that began six months ago.
Occupation forces have spoken of plans to hold a referendum on making the Kherson region a part of Russia and have pressured residents to take Russian citizenship and stop using Ukraine's currency.
Ukraine's presidential office reported "tough battles" going on across practically all of the area and said Ukrainian forces destroyed ammunition depots and all large bridges across the Dnieper River vital to supplying Russian troops.
The Ukrainian military said on Tuesday evening that Russian forces were shelling more than 15 settlements in the Kherson area and resorting to air strikes.
Russian Defence Ministry spokesman Igor Konashenkov maintained that its forces stood up well and that Ukraine lost hundreds of troops, tanks and other armoured vehicles in Monday's action.
Ukrainian independent military analyst Oleh Zhdanov told the Associated Press that "it will be possible to talk about the effectiveness of Ukrainian actions only after large cities are retaken".
He added that Ukrainian forces had breached the first and the second lines of defence in the Kherson region several times in the past "but it didn't bring about results".
"The most important thing is Ukrainian artillery's work on the bridges, which the Russian military can no longer use," Zhdanov said.
The war has turned into a stalemate over the past months, with casualties and destruction mounting and the population bearing the brunt of the suffering during relentless shelling in the east and south.
In other battlefield reports, at least nine civilians were killed in more Russian shelling, Ukrainian officials said, from the Black Sea port of Mykolaiv to the northeastern industrial hub of Kharkiv, where five were killed in the city centre.
The fighting complicates what could be a treacherous trip from Kyiv to Europe's largest nuclear power plant, Zaporizhzhia, by an inspection team from the United Nations' atomic energy agency.
The experts may have to pass through areas of active fighting, with no publicly announced ceasefire, to reach the Russian-occupied plant, where shelling has driven fears of a catastrophe.
Ukraine and Russia have accused each other of shelling the area over and over.
Meanwhile, European Union foreign ministers were likely to agree to suspend a visa facilitation agreement with Russia and make Russian citizens wait longer and pay more for their visas, diplomats said on Tuesday while the bloc remained split over an outright EU travel ban.
Germany and France warned it would be counter-productive to ban ordinary Russian citizens, a move advocated by Ukraine, and the agreement's suspension was a compromise that could be reached at the ministers' two-day meeting in Prague.
with Reuters
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