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"We can now confirm that Russian troops have begun the battle for the Donbas, which they have been preparing for a long time," Mr Zelenskyy said on Telegram.
"A large part of the Russian army is now dedicated to this offensive.
"No matter how many Russian soldiers are brought here, we will fight. We will defend ourselves."
Fighting has intensified in eastern Ukraine after Russia withdrew troops from around the capital Kyiv and refocused its efforts on the Donbas region, which pro-Moscow separatists have partly controlled since 2014.
Shortly before Mr Zelenskyy's address, the Governor of the eastern Luhansk region, Sergiy Gaiday, also announced the start of Russia's much-anticipated attack.
"It's hell. The offensive has begun, the one we've been talking about for weeks," Mr Gaiday said on Facebook. "There's constant fighting in Rubizhne and Popasna, fighting in other peaceful cities.
Russian shelling killed at least eight civilians in eastern Ukraine on Monday, local authorities said.
Mr Gaiday said four people died as they tried to flee the city of Kreminna in Luhansk, which Russian forces captured on Monday.
In the neighbouring region of Donetsk, Governor Pavlo Kyrylenko said four other civilians died in Russian bombardment.
A large Russian offensive in the Donbas region had been predicted by Ukraine's army for weeks.
Russian President Vladimir Putin said he launched the military operation to save the region's Russian-speaking population from a "genocide" carried out by a "neo-Nazi" Kyiv regime.
Mr Putin recognised the independence of two self-proclaimed separatist republics in Donetsk and Luhansk shortly before the full-scale Russian invasion began on February 24.