Captain Sviatoslav Palamar, a deputy commander of the Azov Regiment, said in a video address that both sides were respecting a local ceasefire.
"A special rescue operation is being carried out by the Azov Regiment. We are getting civilians out of the rubble with ropes - it's the elderly, women and children," he said, urging that all civilians, as well as an unspecified number of wounded soldiers, be allowed to leave.
The Azov Regiment began as a battalion of mostly far-right volunteers after Russian-backed separatists seized parts of the eastern Donbas region in 2014. Kyiv later made it a unit of the National Guard and said it was reformed. Russia says destroying the group is a key war aim.
Palamar said he is hopeful they will evacuate all of the civilians and also called for the safe passage for wounded soldiers.
Earlier on Saturday, Russia's TASS news agency, said 25 civilians including six children, had left the territory of the Azovstal steel plant.