The European Union is the only organization at the moment that exhibits a high level of integration amongst its member nation states, but is it the world's first supranational type organization? Are there examples of nation states in the past banding together in an EU like fashion?
International Telegraph Union 1865 as a result of the International Telegraph Convention predates the Universal Postal Union.
Attempts to claim the German Empire or the Catholic Church as supranational organisations flounder on the concept of "nationality" post dating the Westphalian state.
The Latin League was founded in 7th century B.C. by a set of Italian states. The capital city was Alba Longa.
Delian League was founded in 5th century BC
Peloponnesian League was formed between 6 and 4th centuries BC
League of Corinth was formed during the winter of 338 BC/337 BC
Achaean League existed between 280 BC and 146 BC
In central, barbarian, Europe it was common to conduct inter-tribal congresses. Over several such congresses Julius Caesar himself presided.
Late Roman Senate can be considered a supranational organization because the kings and chives of allied countries were allowed to participate.
Catholic Church was founded in 1st cantury AD and soon became a supranational organization with college of cardinals composed of representatives of different countres. The encyclicals of the Pope were mandatory to all Catholic states.
Holy Roman Empire was a supranational organization that can be seen as a prototype of the European Union.
Hanseatic league was created in 13th century.
The earliest political body designed to harness the power of multiple independent sovereign states for trade and military purposes we have a historical record for appears to be the Awussa League.
The Assuwa League was an alliance of city-states and kingdoms formed to oppose Hittite influence in Anatolia, dating to just before the 14th century BCE.
The US Articles of Confederation may be an example. Each colony/state were considered sovereign under it while they ceded some powers, such as common defense, to the federal Congress. Since the colonies were never recognized by other powers as separate nations but as part of the US, it may not fit your question though.
There have been a number of international organizations meant to coordinate certain activities between nations, such as the previously mentioned International Telegraph Union and the International Union of Railways, but these only affected a narrow area.
As Samuel Russell mentioned, the Catholic Church might or might not be considered such an organization. While not precisely like the EU, it did wield considerable political influence over nations for centuries before the modern nation-state era.