DONETSK, Ukraine (REUTERS) - Ukrainian forces fought with separatists in the city of Donetsk for a second day on Tuesday after inflicting heavy losses on the rebels and the government vowed to press on with a military offensive "until not a single terrorist" was left.

Pro-Russian rebels said more than 50 of their fighters had been killed. The mayor of Donetsk, an industrial hub in eastern Ukraine, said the death toll in the clashes which erupted on Monday stood at 40, including two civilians.

A Reuters correspondent counted 20 bodies in combat fatigues in one room of a hospital morgue, some of them missing limbs.

Ukraine used air strikes and a paratroop assault on Monday to clear rebels from Donetsk's international airport and had pushed the separatists out of the complex by the end of the day.

But shooting continued through the night and on Tuesday the road to the airport bore signs of fighting overnight and heavy machinegun fire could be heard in the distance in mid-morning.

"The airport is completely under control," Interior Minister Arsen Avakov told journalists in the capital Kiev.

"The adversary suffered heavy losses. We have no losses," he added.

First Deputy Prime Minister Vitaly Yarema said on the margins of a government meeting: "We'll continue the anti-terrorist operation until not a single terrorist remains on the territory of Ukraine.

"We have 29 or 30 dead on our side but that is not the final count yet," said Dima Gau, a rebel representative.

Donetsk mayor Alexander Lukyanchenko said 40 people had been killed in the past day, 38 of the bodies being of those involved in fighting around the airport.

The prime minister of the rebels' self-styled Donetsk People's Republic (DNR), Mr Alexander Borodai, was quoted by Russia's RIA news agency as telling a news conference in the city: "We have lost more than 50 volunteers."

OFFENSIVE

Ukraine launched the fresh offensive against the rebels, who have occupied strategic points in Donetsk and other towns in the east, after Ukraine's newly elected president rejected any talks with "terrorists".

Mr Petro Poroshenko, a 48-year-old billionaire oligarch, won overwhelming support in Sunday's election that many hope will draw a line under six months of upheaval. He said a robust military campaign in the east should be able to put down a separatist revolt in "a matter of hours".

On the road to the airport, Reuters journalists saw blood-spattered and bullet-riddled hulks of trucks of the kind the separatist fighters have been using.

Though the rebels backing the self-proclaimed republic had suffered serious losses, the heavy machine gunfire indicated that government forces were still facing resistance.

It was too early to say if the apparent military successes of government forces could break the rebellion in Donetsk, the first of a rash of such revolts across eastern Ukraine after months of turmoil in the capital that led to the ousting of the Moscow-backed president, Viktor Yanukovich.

In Donetsk on Tuesday, a hockey stadium was set ablaze, though firefighters managed to bring it under control.

A Kamaz truck of the type rebels use to ferry their fighters was wrecked on the airport road, covered in blood and completely riddled with bullet holes. Blood had spattered across the entire roadway and even reached a billboard seven metres above.

Outside the city centre, another Kamaz was flipped over, its deck covered in blood.

The mayor reinforced warnings for people to avoid the airport area and said nine schools and two hospitals had been closed, with patients evacuated elsewhere. He reassured people that food and other supplies were still reaching the city.