The troops killed the youth during a violent operation at Larnoo in Kokernag area of the district.
The encounter broke out early morning today after a joint team of the Jammu and Kashmir Police and the Indian Army cordoned off an area in Anantnag's Kokernag forest.
The identities of the slain militants are being ascertained and combing operations are underway, officials said.
The police said that Farhan was a "hardcore stone-pelter turned militant" against whom several cases were registered in Kulgam police stations. Hindustan Times reported that one of the two suspected militants killed in the encounter was 18-year-old Farhan Wani, a Kashmiri teen whose father reportedly took to Facebook in November, appealing his son to return home and leave militancy.
Locals told Kashmir Reader that Farheen was a Class 12 student before he joined Hizb ranks in June previous year.
Director General of Police (DGP) SP Vaid told Kashmir Reader that the youths pelted stones at the army, which opened fire, killing one and injuring another.
The security force later fired at the demonstrators, resulting in injuries to a youth. "The fire was retaliated", he said.
The other injured, Yasir Ahmed Dar, 30, who was hit in the shoulder, is stated to be stable.
Internet services have been suspended in south Kashmir.