The local mayor's office said 97 people were also wounded in the city of Mariupol by rockets that smashed into a packed residential district early in the morning and then again shortly after noon.
Pro-Kremlin rebels announced a major new offensive on Saturday after Grad rocket fire killed at least 30 people in a strategic government-held Ukrainian port linking rebel territory with Russian-occupied Crimea.
"Obviously, everyone in the city is very scared. The rebels have already seized the airport. And now they are starting to destroy Mariupol itself," city native Eduard told AFP.
A towering cloud of grey smoke billowed over homes and a row of high-rise apartment buildings while fire brigades scrambled to put out blazes sparked by the heavy shells.
"Today, we launched an offensive against Mariupol," Russia's RIA Novosti news agency quoted Donetsk separatist leader Alexander Zakharchenko as saying.
His deputy earlier denied responsibility for the civilian deaths and Zakharchenko did not refer directly to the rocket fire.
But he called the potential capture of the industrial port "the best tribute possible for all our dead."
Ukrainian Prime Minister Arseniy Yatsenyuk immediately asked the UN Security Council to censure Russia for allegedly spearheading the militants' advance on the biggest pro-Kiev city left standing in the shattered war zone.
Ukrainian President Petro Poroshenko vowed in a separate statement to deliver a "full victory" over the rebels.
"We are for peace but also accept the enemy's challenge. We will defend our motherland the way real patriots do -- until a full victory," the pro-Western leader said in a statement.
EU foreign policy chief Federica Mogherini warned in Brussels that the latest escalation "would inevitably lead to a further grave deterioration of relations between the EU and Russia."
The southeastern Sea of Azov city of nearly 500,000 provides a land bridge between guerrilla-held regions to the east and the Black Sea peninsula of Crimea that Russia annexed from Ukraine last March.
A rebel assault on the port in August saw Kiev repel the attack at a heavy cost that prompted President Poroshenko to agree to a September 5 ceasefire.
That truce was followed by still more clashes that killed at least 1,500 people and was ultimately rejected by the rebels on Friday.
The separatist leader of Donetsk said on Friday he was ripping up the September agreement and launching an offensive aimed at seizing eastern lands still controlled by the pro-Western authorities in Kiev.
His announcement came just a day after his men scored their most symbolic victory to date by flushing out Ukrainian troops from a long-disputed airport in Donetsk that Kiev had clung on to since May.
'Fear and panic'
Western diplomats linked that advance to a new infusion of Russian troops -- firmly denied by the Kremlin -- designed to expand separatist holdings before the singing of a final truce and land demarcation agreement.
The August push on Mariupol -- an important industrial port that exports Donetsk coal and steel -- coincided with the release of NATO satellite imagery purporting to show Russian tanks and forces crossing into the war zone.
Moscow denied ever dispatching its units and called Russian soldiers in the area volunteers or troops who were on leave and fighting to protect locals from alleged persecution by nationalists in the new Kiev government.
Ukraine claimed on Monday that Moscow had poured nearly 1,000 more Russian soldiers and dozens of tanks into the southeast in order to secure control over factories and coal mines that could help the rebels build their own state.
"Most regional exports and imports go through Mariupol. And it has two of Ukraine’s largest smelters that underpin the entire economy of the southeast," said Oleksandr Sushko of Kiev's Institute for Euro-Atlantic Cooperation.
"This attack is designed to sow fear and panic across Ukraine," said the analyst.
Putin on Friday rejected the charges and blamed Kiev for the latest surge in deaths.
"Artillery is being used, rocket launchers and aviation, and it is used indiscriminantly and over densely populated areas," Putin said.
International monitors said that in recent weeks the average daily death toll was nearly 30 -- a level last seen at the very height of a nine-month war that has claimed more than 5,000 lives.
Moscow concedes that militias have recently gained more ground than allowed under the September truce terms.
Russian Foreign Minister Sergei Lavrov promised during tense talks with his Ukrainian counterpart this week to use Moscow's leverage with the insurgents to rein in their attacks.
Moscow has not yet responded to Zakharchenko's decision to discard peace talks completely and to launch a new campaign.
By Oleksandr Stashevsky and Dmitry Zaks in Kiev (AFP)