Witnesses said the clashes late on Monday started when an Afghan army cadet shouted abuse and spat at a banner carried by a group of Shia Muslims commemorating the slaying of revered leader Imam al-Husayn more than 1,300 years ago.
"After that, the Shia Muslims started throwing stones at them (the cadets)," said Husayn, a Shia Hazara who was at the scene.
"Then armed men from the army school started firing on people, and one person died and another was wounded. All the other 15 were wounded by stones being thrown," he said.
Muhammad Zaki, an eyewitness who was slightly injured by a stone thrown during the clash, gave a similar account.
But Afghan army chief Bismillah Khan denied his cadets had fired on the crowd, and said one person had been crushed to death when Shia Muslims pulled a wall down on top of him.
"After that, the Shia Muslims started throwing stones at them (the cadets)"
Khan also denied that army cadets had provoked the Shia Muslims, saying that in fact they had tried to arrest two people who were hurling insults at the mourners.
On Tuesday, nearly 150 people were killed in bomb attacks on Shia Muslim processions in Iraq. More than 20 people were also killed in Pakistan as Shia Muslims marked the martyrdom of al-Husayn, grandson of Prophet Muhammad.
Violence between Shia Muslims and majority Sunni Muslims is rare in Kabul, but Shia Hazaras have a history of being persecuted elsewhere in Afghanistan.
In the latest incident, 12 Hazaras were killed in January by unidentified gunmen in the southern province of Helmand, an area dominated by Sunni Pashtuns.
Hundreds of Hazaras were slaughtered by the former Taliban government in the northern city of Mazar-i-Sharif in 1998.