In his nightly televised address on Saturday, Mr Zelensky said Ukraine's military is pushing the war "onto the aggressor's territory."
This comes five days after Ukraine began its operation, which has taken Russia by surprise and prompted mass evacuations across both sides of the border.
In Ukraine, the capital Kyiv and in the Sumy region came under Russian air attack in the early hours of Sunday, officials said.
A 35-year-old man and his four-year-old child were killed in Kyiv, Ukraine's emergency services said on Sunday.
Three more were wounded, including a 13-year-old child, according to a statement posted on the Telegram messaging app.
Kyiv mayor Vitali Klitschko posted on Telegram early on Sunday morning that air defence units were "operating" and warned civilians to stay in shelters.
In his Saturday address, Mr Zelensky thanked Ukraine's "warriors", and said he had discussed the operation in Russia with the country's senior military commander Oleksandr Syrsky.
"Ukraine is proving that it can indeed restore justice and ensure the necessary pressure on the aggressor," he added.
Reports say Ukrainian troops are threatening to seize one regional town as they fight more than 10km (six miles) inside Russia - the deepest advance since Moscow launched its full-scale invasion of Ukraine in February 2022.
At least 13 people were injured - two seriously - in Kursk in the early hours of Sunday, the region's acting governor Alexei Smirnov said.
More than 76,000 people have already been evacuated from the border area, according to Russia's state Tass news agency, and Mr Smirnov said on Sunday he had ordered officials to speed up the operation to get civilians to safety.
Russia's National Counter-Terrorism Committee imposed a "counter-terrorism operation" regime across three regions on Friday in response to Ukraine's surprise cross-border incursion.
This means authorities in the border regions of Kursk, Belgorod and Bryansk regions can restrict the movement of people and vehicles and use phone tapping among other measures.
Russia said that up to 1,000 Ukrainian troops, supported by tanks and armoured vehicles, entered the Kursk region on Tuesday morning.
The Ukrainians have since reportedly seized a number of villages, and are also threatening the regional town of Sudzha.
On Friday, a video emerged purportedly showing armed Ukrainian soldiers who claimed to have control over the town, as well as a key Russian gas facility there owned by the Gazprom company.
BBC Verify has now confirmed that the footage was indeed from the Gazprom facility on the north-western outskirts of Sudzha, about 7km from the border with Ukraine. The video alone does not verify the claim that Ukrainian troops have taken the whole town.
Russian military bloggers earlier claimed that the town was in Moscow's hands.
BBC Verify checked and confirmed the location of another video posted online on Friday morning. It shows a 15-vehicle Russian convoy damaged, burned and abandoned on a road through the town of Oktyabrskoe, roughly 38km from the border on the Russian side.
The footage also shows Russian soldiers - some injured, some possibly dead - among the vehicles.
Moscow has since sent reinforcements - including tanks and rocket-launcher systems - to the Kursk region.
In its latest report on Saturday morning, the Russian defence ministry said its troops were "continuing to repel the attempted invasion" of Ukrainian forces.
It claimed that Ukraine's attempts to "break through deep into Russian territory" had been foiled.
The Russian claims have not been independently verified.
On Friday, the UN nuclear agency urged both Russia and Ukraine to "exercise maximum restraint" as the fighting was edging closer to the Kursk nuclear power plant - one of the biggest such facilities in Russia.
International Atomic Energy Agency (IAEA) chief Rafael Grossi said measures must be taken “to avoid a nuclear accident with the potential for serious radiological consequences”.
The power plant is located about 60km north-east of Sudzha.
Jaroslav Lukiv & Sofia Ferreira Santos