Lana Parasayeva, a spokeswoman for the province's government, told the AFP news agency that three members of South Ossetia's security forces were among the dead.
She also said that Georgian forces later opened fire on Tskhinvali with heavy weapons, killing another three civilians and wounding three more.
Eduard Kokoity, South Ossetia's de facto president, told Interfax news agency that "our response to Tbilisi's aggressive actions will be very tough and hard-hitting".
"We reserve the right to strike Georgian cities. We have something that can reach them."
Shota Utiashvili, the Georgian interior ministry spokesman, denied that Georgian positions had fired first.
"The Ossetians opened fire, including with grenade launchers, and the Georgian side only returned fire."
He also said that Georgian police, not military forces, had been involved and that there are no Georgian snipers in South Ossetia.
There were no reports of casualties among Georgian forces, he said.
Fighting in South Ossetia, a small mountainous region that broke away from Georgian control during a war in the early 1990s, often flares up during the summer.
Tensions over South Ossetia and Abkhazia, another breakaway Georgian region, have soared in recent months since Russia announced it was establishing formal ties with the separatists.
Unrest increased further last month with a series of bombings in Abkhazia, which the Abkhaz leadership blamed on Georgia, along with Moscow's admission that it had sent military jets on flights over South Ossetia.