The storm has lashed swathes of the south of the country, as well as the Moscow-annexed Crimea peninsula, occupied Ukraine and southern Russia.
"Ten people died in Ukraine due to bad weather. Twenty three people, including children, were injured," the interior ministry said on social media.
Ukraine's energy grid and rescue services have been over-extended by Russia's nearly two-year invasion.
The ministry said the hardest hit area was the southern Odesa region, where emergency services said they had provided assistance to nearly 2,500 people.
Five of those killed were in the Odesa region, officials said.
The neighbouring Mikolaiv region was also badly affected, as were the regions of Kirovograd, Kyiv and Kharkiv.
In Crimea, wind speeds had reached over 140 kilometres (about 90 miles) per hour in some places.
Moscow reported on Monday that four people were killed by the extreme weather in Crimea and Russia, which also left almost two million affected by power cuts.
The day after the storm, the Kremlin-backed head of Crimea, Sergei Aksyonov, said some 93,000 people were still without power and around 245 villages had water supply problems.
He promised to restore power.
Aksyonov also said he had introduced an "emergency regime" in 10 municipalities across the Black Sea peninsula.
"The extent of the damage caused by the bad weather is still being assessed," Aksyonov said on social media.
He added that he had briefed Russian President Vladimir Putin on the situation.
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