The battle for the main transport hub in the rebel-held east erupted yesterday, in a sharp escalation of the conflict after president-elect Petro Poroshenko vowed to take a tough stand against the "terrorists".
"The airport is under our full control. The enemy suffered heavy losses. We have none," Interior Minister Arsen Avakov said in a statement.
He said however that the military was continuing its operation at the airport, and AFP journalists reported hearing explosions and heavy gunfire during the morning.
Donetsk mayor Oleksandr Lukyanchenko told reporters that two civilians and 38 combatants had died, while rebel leaders suggested that the toll among their ranks could be higher.
An AFP correspondent reported seeing body parts and blood splattered near a bullet-riddled truck on the airport road, where makeshift blockades had been set up with dumper trucks and piles of tyres.
In Moscow, Russian Foreign Minister Sergei Lavrov called for all sides to halt the violence.
"The number one task for the Kiev authorities and the test of their durability, taking into consideration the results of the presidential election is bringing an immediate end to the use of the army against the public and an end to any violence by all sides," Lavrov said.
Ukrainian combat jets and helicopter gunships had attacked the Donetsk airport terminal yesterday after it was seized by scores of separatist gunmen waging an insurgency against Kiev's rule.
The strikes represented the most forceful action by Kiev in its battle to crush a bloody pro-Moscow rebellion that has raged in the coal and steel belt since early April, threatening to tear apart the former Soviet state.
The fighting flared on the very day that billionaire oligarch Poroshenko was formally declared the winner of Sunday's presidential election with 54 per cent of the vote.
The 48-year-old pledged to work to restore stability in Ukraine after months of turmoil and rebuild relations with Russia while steering the country on a westward path.
He said Ukraine would press on with its offensive against the insurgents who control about a dozen cities and towns, saying he wanted it to be faster and "more efficient".
And he insisted there would be "no talks with terrorists" until they laid down their arms.
"Their goal is to turn Donbass (east Ukraine) into Somalia. I will not let anyone do this to our state."
Sunday's vote was seen as the most important since Ukraine's independence in 1991 as it fights to stay united and avert economic collapse after years of Soviet-era mismanagement and rampant corruption.
But the insurgency, which has already cost at least 150 lives, thwarted polling in much of the east and rebels have defiantly refused to recognise the result.
Russia, threatened with more Western sanctions if it meddled further in Ukraine after its seizure of Crimea in March, had said it was willing to work with the new leader.
"We are ready for pragmatic dialogue, on an equal footing, based on respect for all agreements, in particular in the commercial, economic and gas spheres," Lavrov said Monday. "We respect the result of the choice of the Ukrainian people."
On Tuesday, however, Lavrov said that a visit to Moscow by the new president was "not being considered."
Poroshenko also said he was ready to "engage" with the Russian leadership and was optimistic a meeting could be arranged with President Vladimir Putin soon.
Analysts see Poroshenko's emphatic victory as a first step in trying to restore stability but, said Holger Schmieding of Berenberg Bank: "That does not end the conflict in Ukraine."
Observers with the Organisation for Cooperation and Security in Europe said the election "largely upheld democratic commitments" and provided the new leader with legitimacy.
US President Barack Obama praised "courageous Ukrainians" for voting in the face of the militant threat and said Washington looked forward to working with the new president.
The ballot was called after pro-Russian president Viktor Yanukovych -- his corruption-stained regime long a source of discontent -- was ousted in February in the bloody climax of months of protests sparked by his rejection of closer EU ties.
Putin responded by seizing Crimea and threatening to invade the rest of Ukraine to "protect" the country's ethnic Russian community.
But Russia said last week it had started withdrawing from Ukraine's border around 40,000 soldiers whose presence had raised deep Western suspicions.