"He was one of the wanted commanders," a police spokesman said, adding three soldiers and the owner of the house were injured in the fighting.
Indian troops killed three more fighters in two separate skirmishes in neighbouring Shopian and Bijbehara districts on Tuesday, police said.
The region's chief minister meanwhile praised troops for helping subdue "terrorism" in Kashmir during a high-level security review in the summer capital, Srinagar.
"He [Azad] complimented the army and security forces for their commendable role in controlling terrorism, restoring peace and protecting the lives of the people," an official statement said.
Indian forces are seeking to suppress a separatist revolt that began in 1989 and has claimed at least 42,000 lives by official count.
Kashmiris frequently voice resentment at the presence of an estimated 500,000 Indian security forces in the region who are battling the fighters.