TBILISI, Georgia — The Georgian authorities said Wednesday that their forces had engaged in a fierce gun battle to free five Georgian villagers being held by a band of heavily armed militants who had crossed over the mountainous border from the Russian republic of Dagestan. Three Georgian officers and at least 10 militants were killed, officials said.

Relations between Russia and Georgia have remained tense since the two countries fought a brief war over disputed territory in 2008, and the militant incursion threatened to worsen the already troubled relationship.

President Mikheil Saakashvili of Georgia said Wednesday night that Russia was responsible for “exporting its disorder.” Speaking at his residence in Kvareli, with the Caucasus mountain range behind him, Mr. Saakashvili said: “Disorder and instability behind these mountains will stay there. We will do everything for this.”

Russia has long battled Muslim separatists in Dagestan, as it has in Chechnya and elsewhere in the North Caucasus, and violence often flares in the region. On Tuesday, at least 14 people were killed in two separate attacks in Dagestan, including a highly respected Muslim cleric, Said Afandi Atsayev.