Several suspected members of a terrorist group "refused to surrender and opened fire from automatic rifles and threw hand grenades on counter-terrorist units," Georgia's security service said in a statement.
One soldier was hospitalised with gunshot wounds, the service's spokeswoman Nino Giorgobiani said in a televised statement.
"One suspect has been arrested and a special operation is under way to detain several others who, according to a preliminary information, are not Georgian nationals and are members of a terrorist group," Giorgobiani said.
Rustavi-2 television station reported that the wounded soldier had died during surgery.
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Georgia has no recent history of major terror attacks.
Some 50 Georgians are believed to be fighting currently alongside the Islamic State extremists in Syria and Iraq, Georgian officials have said.
They are mostly ethnic-Chechen Muslim minority residents of a single valley in the country's north-east, which has developed a reputation as a jihadist hotbed.
Ahead of and immediately after the 2008 Russia-Georgia war Tbilisi accused Russian special services of staging terror attacks against civilians and police, including planting an explosive device outside the US embassy in the capital.
Rustavi-2 aired footage of a burning apartment in a multi-storey residential building and dozens of camouflaged soldiers and an armoured vehicle being deployed at the site.