Well, it depends on how accurate you want this battle account to be. Oldest I can think of is an inscription found in Nippur saying:

Sargon, the king of Agade, the King of the Land, laid waste the city Uruk, destroyed its wall; fought with the men of Uruk, conquered them; fought with Lugalzaggesi, the king of Uruk, took him prisoner and brought him in a neck stock to [Nippur].

It is the first text I can think of that provides basic information on who fought who in a battle, who won and what was the fate of the defeated. Sources predating that usually only report fact and a winner of a war if anything.

I can't find the proper source for the date of the inscription itself (it's fucking ridiculous how many important 50+ year old academic books aren't digitalized yet), or its original neither photographed nor transcribed, but said battle happened c. 2270. More or less since unlike Assyrian chronology based on eclipse in 763 and their long lists of limmu (yearly officials) early Babylonian chronology is much less certain and it might have actually happened 56 years earlier.

I'm not an expert on Egypt by any stretch of an imagination tho, so it's possible that there is something more suitable from I to VI dynasty. If so, someone please correct me.

Edit:

Ok, managed to find good old Kramer's "The Sumerians" 1964, online.

See page 324 (354 of the PDF), inscription 29, but also 30. Sorry for not copy/pasting it here, but it get's messy.

Whole book is quite dated, but translations are most likely solid. Still can't find original nor transcript to check it myself.