Falklands power grab: Negotiations demanded 'as soon as possible' on future of islands
Argentine Foreign Minister Felipe Solá thanked "the countries of the hemisphere for their continued solidarity".
Mr Sola said: "The recovery of the full exercise of sovereignty over the Malvinas, South Georgia and the South Sandwich Islands and the surrounding maritime spaces, respecting the way of life of its inhabitants and in accordance with international law, is a permanent and inalienable objective that is enshrined in the National Constitution and constitutes a State Policy."
Latin American countries such as Peru, Brazil, Chile, El Salvador, Guatemala, Panama, Nicaragua, Costa Rica, Dominican Republic, Suriname and Uruguay all showed their support in the assembly to the claim for sovereignty.
Daniel Filmus, secretary of the Foreign Ministry of Malvinas, Antarctica and South Atlantic, said: "The support on the Malvinas issue adopted by the OAS as a clear sign that it is a regional cause, not only of Argentina."
Mr Filmus pointed to new legislation promoted by Argentine President Alberto Fernández which saw the creation of a National Affairs Council relating to the Falklands.
He described the council as "plural body that will have the objective of drawing up and sustaining state policies in the medium and long term".
He said: "This year our country achieved a historic progress to consolidate sovereignty over the islands after the National Congress unanimously sanctioned the law."
Mr Solá was also accompanied by Secretary of Foreign Affairs Pablo Tettamanti, and the Chief of Staff of the Foreign Ministry, Guillermo Justo Chaves.
Speaking ahead of the two-day summit in the US capital, Mr Chaves branded the UK's sovereignty over the Falklands illegal.
He said Argentina's claims over the islands was "a matter of national identity" and that his government was working "with all diplomatic mechanisms to bring it back" under the country's flag.
Mr Chaves said Argentina's claim on the islands should not be subject to "passionate or rational discussion" and insisted it was an "illegally usurped territory".
Mr Chaves said: "The previous government hindered the Malvinas cause after Foradori-Duncan agreement that generated an exchange of technical and scientific information with serious damage to Argentina and its international relations."
The 2016 accord set out areas of closer co-operation for the two countries through increasing trade links, uncovering new investment opportunities, strengthening cultural ties, tackling corruption and organised crime, and increasing links in science and technology.
The Foreign Office insisted the statement did not affect the sovereignty of the islands and the UK remains absolutely clear in its support of the rights of the islanders.
But the Falklands has been steadily climbing back up the political agenda since Mr Macri was defeated by Mr Fernandez in last year's presidential elections and was central to Argentina's presentation to the United National General Assembly earlier this year.
More recently, Central American countries have declared their support for Argentine sovereignty over the Falkland Islands at the UN.
All eight members of the States of the Central American Integration System, SICA, were represented by Nicaragua at the UN Special Decolonisation Committee on the issue of the Falklands.
The eight members are, Belize, Costa Rica, El Salvador, Guatemala, Honduras, Nicaragua, Panama, and the Dominican Republic.
All members agreed that Argentina should be given sovereignty over the islands.
Ambassador and permanent representative of Nicaragua to the United Nations Jaime Hermida Castillo expressed his support.
He stressed the governments of the Argentine Republic and the UK should resume negotiations in order to find a peaceful and definitive solution to the dispute.