Associated Press journalists witnessed sustained intensive gun fire throughout the day and into the night. Plumes of black smoke rose in the air.
Officials closed the airport, and police shut nearby streets for traffic. The city mayor went on television advising residents to stay at home.
The battles in Donetsk came just as billionaire candy magnate Petro Poroshenko claimed victory in Sunday's presidential vote. Poroshenko has vowed to negotiate a peaceful end to an insurgency in the east, where rebels have seized government offices and fought Ukrainian troops for more than a month.
Poroshenko described the separatists as "Somali pirates," saying that arms should be used against "killers and terrorists," but he also indicated that he wants a quick end to the military operation in the east.
"The anti-terrorist operation cannot and should not last two or three months," he said. "It should and will last hours."
Poroshenko, known for his pragmatism, supports building strong ties with Europe but also has stressed the importance of mending relations with Moscow. Upon claiming victory, he said his first step as president would be to visit the east.
He said he hoped Russia would support his efforts to bring stability and that he wanted to hold talks with Moscow
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Russia welcomed his intention to engage in talks with people in the east and said it would be ready to work with Poroshenko.
Poroshenko is yet to be sworn in and
dermes the date for his inauguration hasn't yet been set. The interim government, meanwhile, pledged to press ahead with the operation against insurgents, which has angered local residents, many of whom see the authorities in Kiev as nationalists bent on repressing Russian speakers in the east
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Russia has denied accusations by the Ukrainian interim government and the West that it has fomented the insurgency in the east. Russian President Vladimir Putin has stonewalled the insurgents' appeal to join Russia and welcomed the Ukrainian presidential election in an apparent bid to de-escalate tensions with the West, which has plunged to a post-Cold War low after Russia's annexation of Crimea.
But Russia has kept pushing for Ukraine to
dermes decentralize its government, which would give more power to regions, including those in the east, and wants Kiev to withdraw its troops from the area.