Ukraine's interior ministry says the military is now in full control of the airport in the eastern city of Donetsk after a day of bloody clashes.
More than 30 pro-Russia separatists were reported killed after an attempt to seize the airport early on Monday.
New President Petro Poroshenko vowed to tackle the eastern uprising in hours, not months. Russia has called for an immediate end to military action.
Meanwhile, the OSCE says it has lost contact with a monitoring team.
The Organization for Security and Co-operation in Europe said four of its monitors were on a routine mission east of Donetsk when they were stopped at a checkpoint at about 18:00 (16:00 GMT) on Monday.
A spokesman told the BBC the monitors, all male, were Turkish, Swiss, Estonian and Danish. Danish trade minister Mogens Jensen said it was believed they were being held by armed separatists.
In April, seven international military observers linked to the OSCE were held captive in eastern Ukraine for a week.
US President Barack Obama had telephoned Mr Poroschenko on Tuesday to congratulate him on his victory in Sunday's elections and offer him "the full support of the United States", said the White House.
Ukraine's Interior Minister Arsen Avakov said the airport was "under our full control" but operations were continuing.
Sporadic gun and artillery fire could be heard into the afternoon.
At the scene: Mark Lowen in Donetsk
Back in February, when pro-Russia separatists launched their incursion in Crimea - which led to Moscow's annexation of the peninsula a month later - the airport was the first key installation they took control of, so Kiev was committed to clamping down on that.
Ukraine's new president gave a speech on Monday saying he would not negotiate with "terrorists", as he put it - and clearly you see the impact of that around Donetsk airport.
A local coroner told the BBC he had seen 37 bodies, among them two civilians.
Photographs from inside Donetsk morgue showed more than a dozen bodies in military fatigues piled up.
A representative of the self-proclaimed Donetsk People's Republic separatists said many fighters had died when a truck taking them to hospital came under government fire.
Other reports of the death toll vary: Donetsk mayor Alexander Lukyanchenko said 40 people had been killed in the past day, while other rebel leaders said the final toll could rise above 100.
The separatists stormed the Sergei Prokofiev Donetsk airport early on Monday, sparking a swift air and land assault by the Ukrainian military.
The attempt to seize the airport may have been intended to prevent Mr Poroshenko from travelling there after he said his first trip would be to visit the restive east.
Mr Poroshenko, a 48-year-old billionaire and former foreign minister, has vowed east Ukraine will not be "turned into Somalia".
However, he has also said he wants to talk to Russia to end the crisis.
Moscow had said it would accept the election results and engage in dialogue with the winner.
But Russian Foreign Minister Sergei Lavrov said on Tuesday that a visit to Russia by Mr Poroshenko was "not being considered".
He again called on Mr Poroshenko to stop military operations in eastern Ukraine immediately and implement a roadmap for peace negotiated in Geneva on 17 April.
Donetsk and Luhansk declared independence after Russia annexed the Crimean peninsula in the wake of the removal of Ukraine's pro-Moscow President Viktor Yanukovych.