Authorities locked down central Kabul, blocking major intersections with stacked shipping containers as the protesters sought to march on the presidential palace, waving flags and chanting slogans such as "death to discrimination".
"(President) Ashraf Ghani and (Chief Executive) Abdullah Abdullah, return us our rights," chanted protesters, which included a large number of women.
The 500-kilovolt TUTAP power line, which would connect the Central Asian nations of Turkmenistan, Uzbekistan and Tajikistan with electricity-hungry Afghanistan and Pakistan, is seen as a crucial infrastructure project.
But leaders from the minority group are demanding that the line be routed through Bamiyan, one of the most deprived areas of Afghanistan with a large Hazara population.
The multi-million-dollar line was originally set to pass through the central province but the government rerouted it through the mountainous Salang pass north of Kabul, saying the shorter route would speed up the project and save millions of dollars.
Hazara leaders in the ethnically divided nation lashed out at the Pashtun president, calling the decision prejudiced against the Hazaras, a community that has suffered a long history of oppression.
Saturday's protest follows a similar demonstration in May, which drew tens of thousands of people.
"We will continue our peaceful campaign until the government says 'yes' to our righteous demands," read one of the banners at Saturday's protest.
The protest came despite calls by some Hazara leaders in the government to pursue talks with the government over the issue rather than demonstrations, which paralyse normal life in the capital.
The dispute, which highlights the challenges of modernising the country, threatens to overshadow the TUTAP project, which is due to be implemented by 2018 and could help ease crippling nationwide blackouts.
The demonstration spotlights the war-torn nation's turbulent politics. It follows one of the biggest anti-government rallies for years last November, which was galvanised by the beheading of a group of Hazaras.
The three million-strong Hazara community has been persecuted for decades, with thousands killed in the late 1990s by Al-Qaeda and the mainly Pashtun Sunni Taliban.
Demonstrators from Afghanistan's Hazara minority attend a protest in Kabul, Afghanistan
Thousands of minority Shia Hazaras demonstrated in Kabul Saturday, demanding that a key power transmission line pass through their electricity-starved province, in the second major protest over the issue this year.
