The fighting erupted on Friday as the UN Security Council held a late-night session to discuss the conflict.
The separatists, who have controlled South Ossetia since the early 1990s, vowed to repel the attack without calling for help from Russia.
"At about 3:30am (2330 GMT on Thursday) a tank attack was launched on the southern outskirts of Tskhinvali," Interfax quoted Eduard Kokoity, the South Ossetian separatist leader, as saying.
"A heavy battle is under way."
| Kokoity, the South Ossetian rebel leader, described the attacks as "perfidious" [AFP]
Kokoity described the attack on Tskhinvali as a "perfidious and vile" act by Mikheil Saakashvili, the Georgian president.
At least 15 civilians were reported killed as Tskhinvali came under heavy Georgian shelling.
A South Ossetian government website provided a description of what it called Georgia's tank attack on Tskhinvali.
Ria-Novosti news agency quoted an official speaking for South Ossetia's ministry of emergency situations as saying: "Tskhinvali is being shot at by mortar and heavy weapons from the Georgian villages of Nikozi and Ergneti and some houses are burning."
Georgian fighter jets bombed separatist positions overnight as well, local authorities and Russian media sources said on Friday.
Jonah Hull, reporting for Al Jazeera from the South Ossetian border with Georgia, said there had been a fairly sustained aerial bombardment of villages thought to be South Ossetian outposts.
But he said the situation was calm after the bombing.
"There is a lull in the bombardment that has been ongoing overnight and throughout the morning," he said.
"I spoke to the Georgian interiror ministry spokesman who said the lull was designed to allow civilians to escape from the city of Tskhinvali. We have no idea at this point how many civilians are trying to flee or indeed where they are going."
Hull said he had seen jets in the air that "look like Georgian jets and there are reports coming in from the Georgian side that Russian jets have been in the air over Tskhinvali and have dropped bombs on the Georgian positions".
But he added that he could not confirm these reports.
Villages taken over
Temur Iakobashvili, Georgia's reintegration minister, confirmed that "Tskhinvali is surrounded by Georgian forces", saying that government forces had taken control of five Ossetian villages loyal to separatists.
"We are forced to restore constitutional order in the whole region," General Mamuka Kurashvili, the head of Georgian peacekeepers in the region, said in comments broadcast on Georgia's Rustavi-2 television network.
"Despite our call for peace and a unilateral ceasefire, separatists continued the shelling of Georgian villages."
Lado Gurgenidze, Georgia's prime minister, said on Friday that Georgia will continue its military operation in South Ossetia until a "durable peace" is reached.
"These actions will continue until we manage to reach a durable peace because people are still in danger," he said.
"As soon as a durable peace takes hold we need to move forward with dialogue and peaceful negotiations."
The Security Council began an emergency meeting late on Thursday at Russia's request on the escalating crisis.
Al Jazeera's John Terrett, reporting from New York at the UN, said the meeting to discuss the conflict was called for by Russians and that they wanted to get the issue on the international agenda.
"We want to try and end the violence which has been escalating," he quoted the Russian delegation as saying.
Moscow had urged the session, saying the world community ought to avert "massive bloodshed.
"It is not too late to avert massive bloodshed and new victims," the Russian foreign ministry said in a late-night statement.
"We hope our foreign partners will remain impartial in this difficult moment, when the fate of hundreds of thousands people is decided," it said. "The Georgian leadership should ... return to civilised ways of solving complicated problems."
Dmitry Medvedev, the Russian president, has been holding consultations with aides on Moscow's strategy in the South Ossetian conflict, the Kremlin said.
"A complex of measures is now being discussed under the guidance of Dmitry Medvedev aimed at restoring peace in South Ossetia, defending the local civilian population within the peacekeeping mandate we have," a Kremlin spokesman said.
South Ossetia and Abkhazia, another Georgian breakaway province, receive extensive political and financial assistance from Moscow.
Tbilisi has repeatedly accused Russian peacekeepers of supporting the separatists, while Moscow has accused Georgia of planning a full-scale invasion to re-establish control over the region.
Saakashvili had on Thursday offered an immediate halt to heavy fighting, after a week of clashes in which nearly 20 people were killed.
"Let's stop this spiral of violence ... let's resume negotiations," he had said in his televised address.
Saakashvili also reiterated a previous offer of "practically limitless" autonomy for South Ossetia and proposed that Russia could be the guarantor for any deal.
Martin McCauley, a London-based Russia analyst, told Al Jazeera: "You can argue that the president of South Ossetia, who wants independence from Georgia, is deliberately provoking Tbilisi and is trying to suck Russia in."