Vehbi Avuc, governor of the Bingol province, said the soldier was shot dead and eight others were injured when members of the banned Kurdistan Workers Party (PKK) opened fire on them on Sunday evening.
A security official said the three PKK fighters were killed on Monday as troops, backed up by helicopters, pressed their offensive against rebels hiding in the area.
"Operations are continuing," he said.
Last week, two PKK fighters, three soldiers and three village guards were killed in various clashes across Turkey's impoverished, mainly Kurdish, southeast.
Ankara blames the PKK for the death of more than 30,000 people since the group began its armed campaign for a separate Kurdish homeland in southeast Turkey in 1984.
Violence has increased since the PKK called off a unilateral ceasefire in 2004. Every spring, the Turkish army steps up operations in the southeast as the mountain snows melt and rebels based in northern Iraq cross into Turkey.
Ankara has urged US and Iraqi government forces to crack down on the PKK units holed up in the mountains of mainly Kurdish northern Iraq.