(Reuters) - Russia pressed for control of Ukraine's Donbas region, claiming victory in the months-long battle for Mariupol's steel plant and launching a major offensive on the remaining Ukrainian-held territory in the province of Luhansk.

The last Ukrainian forces holed up in Mariupol's smashed Azovstal steelworks surrendered on Friday, Russia's defense ministry said. That ended the most destructive siege of the war.

"The territory of the Azovstal metallurgical plant ... has been completely liberated," the ministry said in a statement, adding that 2,439 defenders had surrendered in the past few days, including 531 in the final group.

Hours earlier, Ukrainian President Volodymyr Zelenskiy said the last defenders at the steelworks had been told by Ukraine's military that they could get out and save their lives. The Ukrainians did not immediately confirm the figures on Azovstal.

Ukraine's General Staff of Armed Forces did not comment on Russia's claim in its morning update on Saturday.

Russia also launched what appeared to be a major assault to seize the last remaining Ukrainian-held territory in Luhansk, one of two southeastern Ukrainian provinces Moscow proclaims as independent states.

Serhiy Gaidai, the governor of the Luhansk region, said in a social media post early on Saturday that Russia was trying to destroy the city of Sievierodonetsk, with fighting taking place on the outskirts of the city.

"Shelling continues from morning to the evening and also throughout the night," Gaidai said in a video post on the Telegram messaging app.

In early hours on Saturday, air raid sirens were going off in much of Ukraine, including in the Kyiv capital region and the southern port of Odesa.

Capturing Luhansk and Donetsk provinces, much of which make up Ukraine's industrial Donbas region, would allow Moscow to claim a victory after announcing last month that this was now its objective.

Despite losing ground elsewhere in recent weeks, Russian forces have advanced on the Luhansk front.

"This will be the critical next few weeks of the conflict," said Mathieu Boulegue, an expert at London's Chatham House think tank. "And it depends on how effective they are at conquering Sievierodonetsk and the lands across it."

The city of Sievierodonetsk and its twin Lysychansk across the Siverskiy Donets river form the eastern part of a Ukrainian-held pocket that Russia has been trying to overrun since mid-April after failing to capture Kyiv.

Ukraine's general staff said on Saturday that Russian forces were preparing to try again to cross the river, after a previous attempt earlier this month led to one of the largest battles in the conflict so far.