Muslim-majority Kashmir has been divided between rivals India and Pakistan since their independence from British rule in 1947 and is home to a long-running insurgency.
The first firefight broke out after Indian troops moved to check "suspicious movement" near Halkan Gali in Anantnag, resulting in the death of two militants.
The Indian Army's Chinar Corps said on social media platform X that a joint operation had been launched in the area.
"Terrorists opened indiscriminate fire," the corps said in its statement. "Troops effectively retaliated, resulting in elimination of two terrorists."
Another suspected militant was killed after a gunfight with police and army soldiers in Srinagar, a senior police official confirmed to AFP.
The clash started Saturday morning in the Khanyar area of the city after police were searching for armed militants.
Firefights between militants and security forces in the densely populated downtown areas of Srinagar have been rare for many years.
At least 500,000 Indian troops are deployed in Kashmir, battling an insurgency that has killed tens of thousands of civilians, soldiers and rebels since 1989.
Last week suspected militants ambushed an army vehicle and killed five people, including three soldiers.
A week before that, gunmen killed seven people near a construction site for a strategic road tunnel to Ladakh, a high-altitude Himalayan region bordering China.
New Delhi regularly blames Pakistan for arming militants and helping them launch attacks, an allegation Islamabad denies.
The army says more than 720 rebels have been killed since Indian Prime Minister Narendra Modi's government cancelled the territory's limited autonomy in 2019.
In early October, Kashmir held its first elections since 2014 for a regional assembly for the territory of some 12 million people.
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