The obvious answer is the ancient Olympic Games, first established in 776 BC (according to legend), when Herakles1, Paeonaeus, Epimedes, Iasius and Idas raced at Olympia to honour Zeus. Olympia has been a site of religious activity since as early as the 10th century BC:
Extensive deposits filled with ash and votive offerings from the sanctuary of Olympia indicate the existence of an early long cult tradition dating back to the 10th century and on. Since no buildings have survived from this early period, we must assume that these earliest offerings were placed directly on the altars, or displayed in open.
The region of Elis organized the first Olympic festival in the 8th century BC. According to the tradition, the Olympic games were first held in 776 BC. They included one single athletic event, the one-stade race, won by Coroebus of Elis, the first victor of the Olympic games we know of. Around 700 BC, the site was subject to major reorganization: the ground was levelled off, and many wells were dug to the east. Changes were made at the northern borders of the sanctuary too. Gradually, the programs of the Olympic festivals expanded to include other athletic events for boys.
Do the ancient Olympic Games count as international though? Participation was restricted to free born Greek males2, the judges were called "Ἑλλανοδίκαι"3 and Alexander I of Macedon had to prove his Hellenic ancestry before being allowed to compete as Herodotus reports in Τερψιχόρη (22):
Now that the men of this family are Greeks, sprung from Perdiccas, as they themselves affirm, is a thing which I can declare of my own knowledge, and which I will hereafter make plainly evident. That they are so has been already adjudged by those who manage the Pan-Hellenic contest at Olympia. For when Alexander wished to contend in the games, and had come to Olympia with no other view, the Greeks who were about to run against him would have excluded him from the contest- saying that Greeks only were allowed to contend, and not barbarians. But Alexander proved himself to be an Argive, and was distinctly adjudged a Greek; after which he entered the lists for the foot-race, and was drawn to run in the first pair. Thus was this matter settled.
Alexander had also proclaimed his Hellenic heritage when acting as an emissary for Mardonius, prior to the Battle of Salamis:
"Men of Athens, that which I am about to say I trust to your honour; and I charge you to keep it secret from all excepting Pausanias, if you would not bring me to destruction. Had I not greatly at heart the common welfare of Greece, I should not have come to tell you; but I am myself a Greek by descent, and I would not willingly see Greece exchange freedom for slavery. Know then that Mardonius and his army cannot obtain favourable omens; had it not been for this, they would have fought with you long ago. Now, however, they have determined to let the victims pass unheeded, and, as soon as day dawns, to engage in battle. Mardonius, I imagine, is afraid that, if he delays, you will increase in number. Make ready then to receive him. Should he however still defer the combat, do you abide where you are; for his provisions will not hold out many more days. If ye prosper in this war, forget not to do something for my freedom; consider the risk I have run, out of zeal for the Greek cause, to acquaint you with what Mardonius intends, and to save you from being surprised by the barbarians. I am Alexander of Macedon."
The key word in both passages is "barbarian", a word that at the time was a colloquial description of non Greeks. The most common theory for the word's origin is the "bar bar" sound that Greeks claimed non Greeks sounded like. This is what "Greek" mostly meant at the time, a common language, and not so much a common ethnicity or nationality, at least not by today's definitions.
While speaking Greek was a requirement for participating in all four Panhellenic Games, the competing city-states were independent and autonomous entities, often at war with each other. The Olympic Truce and the tradition of announcing alliances between city states during the Games are further evidence that, at least for the era's definition, the Games were international.
Interestingly, the first recorded case of a naturalized athlete was in the 100th Games, when Sotades competed as an Ephesian citizen, while having won the δόλιχος long distance race for his birthplace, Crete, in the previous games. Pausanias in Ἑλλάδος περιήγησις reports that Sotades was bribed by the Ephesians to run for them (effectively changing citizenship) and was banished from Crete afterwards.
The Games became decisively international after the Roman conquest of Greece, with Roman Emperors Tiberius and Nero and Armenian Prince Varazdat being recorded as Olympic victors.
1 Not the well known demigod, but an Idaean dactyl.
2 But see: Cynisca
3 Literally "Judges of the Greeks"