The Bosnian constitution was an annex to the Dayton Peace accord that ended the 1992-95 conflict in Bosnia, splitting the territory into two entities, the Serb Republic and the Muslim-Croat Federation linked by a three-part presidency.
The provisions of the constitution had intended to prevent ethnic strife in the wake of the war.
The court "acknowledged that this system, put in place at a time when a fragile ceasefire had been accepted by all parties to the inter-ethnic conflict that had deeply affected the country, pursued the legitimate aim of restoring peace".
But it also noted that Bosnia had committed under an association agreement signed with the European Union in 2008 to bring its electoral rules into line with the European convention on human rights.
It said the current constitution violated the provisions of the convention which prohibits discrimination and upholds the right to free elections.
Bosnia was ordered to pay $28,500 to Finci and $1,500 to Dervo Sejdic, the other plaintiff in the case, in costs and expenses.
In a statement, Minority Rights Group International, a non-governmental rights organisation that assisted Finci, said: "The court's decision is binding on the Bosnian government and is likely to require a constitutional change granting equal rights to political participation for members of smaller minorities."