State Director General of Police S P Vaid said the three terrorists who attacked the BSF camp had been "neutralised".
"We are carrying out search operations inside the premises to rule out the possibility of planting of explosives," he said.
BSF Assistant Sub-Inspector B K Yadav, 50, was killed and three jawans injured when the militants entered the BSF battalion headquarters at Gogoland, adjacent to the airport.
Briefing reporters later, Inspector General of Police (Kashmir range) Muneer Khan said the overground network of the terror group had been identified but did not divulge any further information.
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Asked specifically whether the Jaish-e-Mohammed was behind the attack, Khan said, "Such type of attacks have been carried out by this group only."
He said the terrorists were dressed in combat fatigues and entered the camp from an area adjacent to a residential colony by cutting the fence.
The IG added that three members of the same group were killed in an attack on district police lines in Pulwama on August 26.
"Six to seven terrorists of this group are still at large. We have identified the overground network which is helping these terrorists move around," he said.
Asked if more attacks were likely, he said "we have to be ready to face such attacks" as long as there is militancy and militants in Kashmir.
According to Khan, all security installations were important and it was the preparedness of the security forces which repulsed the attack without causing any collateral damage.
The old airfield, being manned by the Indian Air Force, is also located in the area.
Civilian air operations, which were suspended for nearly three hours in the morning, resumed at 10 am. "I personally went to the airport and ensured that the public does not face any problems in catching the aircraft," Vaid said.
Piecing together the chain of events, police said militants entered the BSF's 182 battalion headquarters and started firing in all directions.
Yadav's body was recovered during the mopping up operations.
The jawan, who joined the border guarding force in 1987, hailed from Bhagalpur in Bihar.
The 182 battalion is entrusted with the security of the runway of the Srinagar airport.
Jaish-e-Mohammad, the Pakistan-based terror outfit militant outfit, has claimed responsibility for the attack.
A caller identifying himself as a spokesman of the outfit told local news gathering agencies that Jaish militants had carried out the attack.