Ukrainian forces had committed to retaking the southern regions, including across Kherson, that quickly fell to Russian forces earlier this year. But this effort has been increased since the start of this week, the MoD said.
Late on Tuesday night, Ukrainian authorities claimed they had retaken four villages in the Russian-controlled region.
Several defences around the Black Sea city of Kherson were breached in a dramatic push by Ukraine.
Kyiv announced the start of the concerted retaking of Kherson on Monday, with a former official in the region saying “the first line of Russian defences on the Kherson front” had been broken.
Sergei Khlan continued on the messaging app Telegram: “Today there was a powerful artillery attack on enemy positions in […] the entire territory of the occupied Kherson region.
“This is the announcement of what we have been waiting for since the spring – it is the beginning of the de-occupation of the Kherson region.”
Ukrainian President Volodymyr Zelensky then addressed Russian soldiers on Tuesday: “The occupiers should know we shall oust them to the border, the line of which has not changed.”
He added: “If they want to survive, it’s time for the Russian military to run away. Go home.”
Kremlin spokesperson Dmitry Peskov, responded to the offensive that Russia would still fulfil its objectives, commenting: “All of our goals will be reached.”
“In line with its doctrine, Russia will likely now attempt to plug the gaps in its line using pre-designated mobile reserve units.
“These will likely include some of those from the Eastern Grouping of Forces.”
Ukraine’s Southern Command said that in the first day of the offensive, Russia had lost 82 soldiers and two tanks.
The Russian ministry countered that Ukraine had sustained losses of 1,200 soldiers.
An advisor to Mr Zelensky said that the Ukrainian offensive was designed to “grind the enemy” down, and would not be over quickly.
Kherson holds a certain symbolic value for the war effort, as it was the first city taken by Russia.