Both sides of the conflict are blaming the other for sparking the fighting.
The battle started when Pakistani forces fired on Indian border posts and three villages in the Ranbir Singh Pura region, Dharmendra Pareek, a top official of India's paramilitary force, told the Wall Street Journal.
A man and his 8-year-old son were killed by the fire and four others were injured.
A Pakistani senior officer said things got started when the Indian border security force "resorted to unprovoked firing" near the city of Sialkot, killing two civilians, a woman and a 60-year-old man.
The Kashmir region has seen fighting since the 1940s between India and Pakistan. The region's administration is split between the countries and two wars have been fought over the territory.
The two countries have largely observed a 2003 cease-fire, though sporadic fighting has taken place since then.