Doubtless, people have blown on split blades of grass since the dawn of time; and that's a primitive type of reed.
According to this article, the Egyptians were using reed instruments ca. 3300 BCE. http://en.wikipedia.org/wiki/Predynastic_Egypt
I imagine that date is pretty reliable, Egyptian chronology is well understood.
I looked at the Wikipedia entries of all the ancient civilisations I could think of, and none of them had anything of interest to say about the history of music; although it seems the Greeks were using reed instruments before 500 BCE.
The Egyptian date is the earliest physical record, but I suspect that reed instruments were invented long before that, probably in several different places. They aren't difficult to make; I used to own a simple double-reed pipe I'd bought in Greece from an old man by the side of the road, who was whittling them for sale to tourists. FWIW, I'd guess that the first ones were made by shepherds - watching sheep graze is a boring job, giving plenty of time to make and play pipes.
If you want fuller information, I suggest that you find a library which holds the Oxford History Of Music, and consult that.