The Afghan Independent Human Rights Commission on Monday said the Joint Electoral Management Body (JEMB) had failed to set up a proper complaint handling procedure.
Most candidates opposing interim President Hamid Karzai condemned Saturday's election as fraudulent and illegitimate.
One of the main charges was that the special ink which was supposed to mark voters' fingers - to prevent them voting twice - could be washed off.
The commission said it "strongly urges the JEMB, the United Nations Assistance Mission in Afghanistan and the international community to establish a transparent and appropriate mechanism to address electoral complaints".
The human rights body said an investigation should look at complaints "coming from both the voters and the candidates to increase the trust of the people in the election process".
It further requested the JEMB make their results known to the public.
But the Organisation for Security and Cooperation in Europe (OSCE), which sent advisers to help monitor the polling, said Saturday's demand by 14 opposition candidates to nullify the election was "unjustified".
Afghanistan's electoral management body is working on a mechanism to investigate complaints about the conduct of the vote, a UN spokesman said.