An unknown number of gunmen opened fire on the two army trucks and a police car travelling through Baramulla district, 60 kilometers (37 miles) west of the region's main city of Srinagar.
"Two soldiers and one police officer were killed and three others -- two soldiers and another police officer were injured," district police superintendent Imtiaz Hussain told AFP.
Security forces launched a major search for the militants who fled the scene, according to the Press Trust of India news agency.
Army convoys have been travelling more frequently at night in recent weeks through the region to avoid encountering protesting residents.
Kashmir has been under curfew since protests erupted over the death last month of a popular young rebel leader, Burhan Wani, in a gunfight with security forces.
More than 60 civilians have been killed in clashes between protesters and security forces, and thousands more injured in the worst violence to hit the Himalayan region since 2010.
Kashmir is split between India and Pakistan along a UN-monitored line of control, but both claim it in full and have fought two wars over its control.
Armed rebels have fought Indian security forces in Kashmir since 1989 for the independence of the region or for it to be made part of Pakistan. The conflict has left tens of thousands, mostly civilians, dead.
Gunmen shot and wounded 10 police paramilitaries Monday who were trying to impose a curfew in Srinagar during Independence Day. One of the troopers later died of his injuries.