Ukrainian officials said their soldiers would withstand the assault, calling it the Battle of the Donbas.
But Russian forces pressed an advance across almost the entire stretch of the eastern front and, hours after its start, seized a frontline city.
In the ruins of Mariupol, the southeastern port that has suffered nearly eight weeks of siege, Russia gave the last Ukrainian defenders holed up in a steel works an ultimatum to surrender by noon.
The deadline passed without word of their fate.
Ukraine's lead negotiator said it was hard to predict when peace talks might resume because of the Mariupol siege and the new offensive.
In a swift response to the intensified assault, US President Joe Biden and other allies discussed increasing military, economic and humanitarian support for Ukraine and ways to hold Russia accountable, the White House said.
Chancellor Olaf Scholz said Germany intended to supply Ukraine with anti-tank and air defence weapons as well as long-range artillery weapons.
UK Prime Minister Boris Johnson also promised more artillery weapons as the conflict moved into a new phase.
Scholz said the allies agreed Russia must not win the war, and that an imposed peace as envisaged by Russian President Vladimir Putin was not acceptable.
The Italian government said there had been "broad consensus on the need to step up pressure on the Kremlin, including by adopting further sanctions, and to increase Moscow's international isolation".
UN Secretary-General Antonio Guterres called for a four-day humanitarian pause in the fighting over the Orthodox Easter weekend to allow civilians to leave areas of conflict and humanitarian aid to be delivered.
Ukraine said the new assault had resulted in the capture of Kreminna, an administrative centre of 18,000 people in Luhansk, one of the two Donbas provinces.
Russian forces were attacking "on all sides," authorities were trying to allow civilians to leave and it was impossible to tally the civilian dead, Luhansk regional governor Serhiy Gaidai said.
President Volodymyr Zelenskiy told Ukrainians in a video address overnight: "No matter how many Russian troops they send there, we will fight. We will defend ourselves."
In Moscow, Russian Foreign Minister Sergei Lavrov confirmed that "another stage of this operation is beginning".
Defence Minister Sergei Shoigu said Russia was "methodically" carrying out its plan to "liberate" Donetsk and Luhansk, provinces which Russia demands Ukraine cede fully to Russian-backed separatists.
Driven back by Ukrainian forces in March from an assault on Kyiv in the north, Russia has instead poured troops into the east for the Donbas offensive.
It has also made long-distance strikes at other targets including the capital.
Ukraine's top security official, Oleksiy Danilov, said Russian forces had tried to break through Ukrainian defences "along almost the entire front line of Donetsk, Luhansk and Kharkiv regions".
Russia now says its aim is to capture the full provinces on the separatists' behalf.
Ukraine has a large force defending northern parts of the Donbas and military experts say Russia aims to cut them off or surround them.
But Russia still needs to keep its troops supplied across kilometres of hostile territory.
Ukraine has counter-attacked near Kharkiv in the rear of Russia's advance, apparently aiming to cut off supply lines.
In Mariupol, scene of the war's heaviest fighting and worst humanitarian catastrophe, a last group of Ukrainian defenders holding out in the Azovstal steel works defied Russian calls to surrender and be spared.
However, about 120 civilians living next to the sprawling plant left via humanitarian corridors, the Interfax news agency reported on Tuesday, quoting Russian state TV.
The pro-Kremlin leader of Chechnya, whose forces have been fighting in Mariupol, predicted troops would capture the plant on Tuesday.
"Obviously, against the backdrop of the Mariupol tragedy, the negotiation process has become even more complicated," Ukraine's lead negotiator Mykhailo Podolyak told Reuters.
Ukraine and Russia have not held face-to-face talks since March 29.
Each side blames the other for their breakdown.
