The fighting first broke out Sunday afternoon when Afghan forces and local militiamen tried to stop Pakistani forces from allegedly establishing a military installation along the disputed border, said Abdul Ghani Musamem, spokesman for the governor of Kunar.
He said the shooting lasted over three hours and also wounded four civilians in the remote Nari district, which lies along the border. He said communication with the area was difficult because of the isolated location.
"Dozens of local people have displaced from the area," he said, adding that reinforcements had arrived in the Do Kalam area of Nari district.
There was no immediate comment from Pakistan.
The two countries often trade accusations that each side is firing across the border, which runs for 2,400 kilometers (1,490 miles), much of it through mountainous terrain.
Pakistani troops are currently building border fences in response to ongoing complaints by Afghanistan over cross-border movement, particularly by militants.
But Kabul remains deeply suspicious of Pakistan's motives. Islamabad had backed the Taliban regime in Afghanistan, and until recently seemed unable or unwilling to go after militant leaders taking refuge inside its borders.
Afghanistan also disputes the exact path of the border, known as the Durand Line, which was drawn by the British in 1893 to mark the edge of their colonial possessions.