Well I don't know if it counts, but many of the territories conquered by Alexander the Great came peacefully through diplomatic appeal due to his reputation, or news of a particlar shocking victory in the past.
For example when he moved into India, the city of Taxila peacefully agreed to an apparently rather friendly alliance rather than face war. Also in India, the city of Nysa was left unmolested due to an appeal on the behalf of the people's chief and 30 of the cities distinguished citizens, who praised Alexander, claimed that the city had been founded by the Greek god Dionysus and gave him soldiers to add to him army.
However, this might not be what you were looking for as this was more peaceful conquering than avoiding war. Source: Arrian's "Campaigns of Alexander", Book 5.
This isn't Pre-Rome, but Rome's intervention in the sixth Syrian war is a good example of avoiding war via diplomacy.
In 168 BC, Antiochus IV Epiphanes of the Seleucid Empire was leading a war against Egypt (his second) and had sent a fleet to capture Cyprus as well. This alarmed the Roman Republic, and so the Senate sent Gaius Popillius Laenas (an aging Senator and former consul) to deliver a message to Antiochus: if he did not withdraw his forces from Egypt and Cyprus, Rome would consider this an act of war. When Anitochus met Laenas while the former was marching toward Alexandria, Antiochus at first sought to make a bid for time by saying he would need to consult with his advisers. At this point, Laenas drew a circle in the sand around Antiochus with his cane and said, "Before you step out of that circle give me a reply to lay before the Senate." After considering for a few moments, Anitochus responded, "I will do what the Senate thinks right." Antiochus withdrew his forces and further war was averted. This event may also be the origin of the phrase "a line in the sand" that comes up a lot in diplomatic discourse.
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Would you consider the paying of tribute under the threat of war to be diplomacy? It was a pretty standard tactic in the Ancient world, and for the most part rulers of minor kingdoms preferred that to open war.
The Athenian Democracy under Pericles might fit the bill depending on your interoperation of the lead up to the Peloponnesian War. He invented the defensive alliance and the trade embargo to protect Athenian interest without violating the terms of a peace treaty with Sparta and provoking war, though it didn't work.
Arthur M. Eckstein looks at international relations in the ancient world. He often states that most classical states couldn't go beyond compellence or coercive diplomacy, do to the speed of communication and the lack of standing diplomatic institutions.
We have a lot of evidence for diplomatic interaction from the Late Bronze Age (c. 1500-1200 BCE) in the Eastern Mediterranean. Letters found at Amarna in Egypt and other places are part of the international correspondence between the kings of Egypt, Babylonia, Assyria, the Hittites and Mitanni. The political situation at the time was a balance of power with a lot of small skirmishes but only very rarely outright war. While the letters don't normally mention conflict or war per se, we have evidence for diplomatic gifts being exchanged and diplomatic marriages being arranged. Of course we can't know how real the threat of war really was or if it was just rhetoric...
The Hittite-Ramesside peace treaty from c. 1259 BCE is an example that ended a conflict and seems to have led to a period of relative peace between the two kingdoms.
Certainly we can look at Pericles' actions to the Spartan invasion of Attica in 446 BC as an example of diplomacy (and/or bribery) over conflict. While war was not prevented in the long run, the peace did weather several crises and postponed large-scale conflict for 13 years. Donald Kagan's lectures at Yale that were released a few years ago gives his very pro-Pericles account. In this case it literally was an army in Athenian territory and ready for total war, who withdraw without bloodshed.
I think it's clear that states from their inception have used many alternatives to war. Diplomacy, tribute, bribes to a few key figures, as well as shrinking the scale of the conflict (assassinations rather than invasions, for instance). Societies, especially literate societies, have a lot of convenient tools to solve problems.