The government attacked rebels with air strikes and paratroopers after a group of pro-Russian separatists captured Donetsk airport.
Helicopters were seen firing on the main terminal, while ground troops are reported to have engaged rebels at the airport and the surrounding urban areas.
Rebels have claimed Donetsk as the capital of their would-be breakaway republic, and the seizure of the airport could prevent the new President from visiting the region.
Russian officials told the Ukrainian government to back off in the eastern regions before it would begin negotiations over the fate of the country. But Poroshenko promised a firmer response to “terrorist” activity.
The chocolate industry tycoon won 54 per cent of the vote in Sunday’s elections, according to preliminary results, giving him an outright majority. However, voters in some areas of the east were prevented from entering polling stations by rebels.
“They want to preserve a bandit state which is held in place by force of arms,” Poroshenko said yesterday.
“These are simply bandits. Nobody in any civilised state will hold negotiations with terrorists.”
The President said he wants the military operation to be”faster and more effective,” saying “the anti-terrorist operation should not last two or three months. It should last for a matter of hours.”
Fighting broke out in the weeks after Russia seized the Crimea region of Ukraine, which has now been annexed by Moscow.
But Poroshenko is keen to hold on to the other eastern provinces, which have a minority of Russian speakers.
In recent weeks Russia’s Vladimir Putin has referred to the territories as “New Russia”. His regime has been accused of stirring up the opposition in the east to Kiev’s government.