Ukrainian troops have launched air attacks to regain control of Donetsk airport from pro-Russia rebels.
A fierce gun battle ensued, and smoke can be seen rising from the area.
The fighting comes as the man tipped to become Ukraine's new President, Petro Poroshenko, said he wanted to talk to Russia to end the crisis.
Russia's Foreign Minister Sergei Lavrov said Moscow was "open to dialogue" with Mr Poroshenko but military action against separatists must end.
Mr Poroshenko said he hoped to meet Russian leaders early next month, after a trip to Poland where he will meet the US president and EU leaders.
However, he warned he would take a tough line on armed militiamen.
Petro Poroshenko: "Without Russia it would be much less effective or almost impossible to speak about the security in the whole region"
He said: "Their goal is to turn Donbass [east Ukraine] into Somalia. I will not let anyone do this to our state and I hope that Russia will support my approach."
Meanwhile election observers said Sunday's vote was a genuine one that largely met international standards.
The mission from the Organisation for Security and Co-operation in Europe (OSCE) also said it gave the new president "legitimacy" to open a dialogue with separatists in the east.
Welcoming the election as "a major step", the EU said it was looking forward "to further concrete steps on this constructive path, including the use of leverage on armed groups to de-escalate the situation on the ground".
Mr Poroshenko, 48, currently has about 54% of the vote, with 75% of the ballots counted, and would not need a run-off. Former Prime Minister Yulia Tymoshenko is a distant second on 13%.
Media reaction to election
"Has Ukraine been reborn?" asks the Ukrainian edition ofKomsomolskaya Pravda. It says the election "will have a special place in the history books". But the paper notes that Crimeans did not vote and the voting in Donbass and Luhansk - formerly Ukraine's "electoral core", proceeded with "great difficulty".
"We have survived," proclaims popular Ukrainian daily Segodnya.
Claims in the Russian media that Ukraine is overrun by extremists and neo-Nazis are ridiculed by some social media commentators. The two right-wing candidates - Dmytro Yarosh and Oleh Tyahnybok - polled less than 2% in total.
Prominent Ukrainian journalist Mustafa Nayyem proclaims an "epic fail" of Russian propaganda. "Vladimir Putin won't sleep tonight," he predicts.
Pro-Russia rebels stormed Sergei Prokofiev Donetsk airport on Monday after it became clear that Mr Poroshenko was heading for outright victory in Ukraine's presidential election.
Fighter jets circled over the airport as attack helicopters fired rockets at the building. There are no confirmed reports of casualties.
Ukraine's interim government is engaged in an offensive in the east to quash the uprising that has left scores dead.
There is hope that the Ukrainian election will lead to a de-escalation of the crisis
Pro-Russian separatists severely disrupted voting there. No polling stations were open in Donetsk city, and across the region.
On Monday, Mr Poroshenko said "the anti-terrorist operation cannot and should not last two or three months".
"It should and will last hours."
Addressing reporters in Moscow, the Russian foreign minister said continuing military operations against the separatists would be a "colossal mistake".
Mr Lavrov said: Russia was "ready for dialogue with Kiev's representatives, with Petro Poroshenko" and EU and US mediation were not needed.
He said: "As our president [Vladimir Putin] has said, we shall treat the results of the expression of will of the Ukrainian people with respect."
Sunday's election came three months after pro-Moscow President Viktor Yanukovych fled Kiev amid bloody street protests and calls for closer ties with the EU.
Since then, Russia has annexed the Crimean peninsula in southern Ukraine and armed separatists in the eastern provinces of Luhansk and Donetsk have declared independence.
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