The shooting began Sunday when government forces cordoned off a village on a tip-off that armed militants were hiding there.
“Both the militants were local residents, one of them had recently joined LeT,” said inspector general of police Javid Gillani, referring to Lashkar-e-Taiba, a banned Pakistani group.
Gillani said a stray bullet killed the civilian in Redwani village, 65 kilometres (40 miles) south of the main city of Srinagar.
A police officer on the scene said hundreds of residents from nearby villages came out to protest as troops cordoned off the village, throwing rocks at the them and shouting “we want freedom”.
“They were trying to help the trapped militants escape,” said the police officer, speaking on condition of anonymity.
Police fired tear gas to disperse the protesters, he added.
Army spokesman N N Joshi said two soldiers were being treated in hospital for gunshot wounds.
Kashmir has been divided between India and Pakistan since the two countries gained independence from Britain in 1947. Both countries claim the disputed territory in its entirety.
Since 1989 several rebel groups have been fighting hundreds of thousands of Indian government forces deployed in the region, for independence or a merger of the Himalayan territory with Pakistan.
Tens of thousands of people, mostly civilians, have died in the fighting. -AFP