It actually isn't entirely certain. They first show up in artwork from the Mediterranean around the late second and early third century CE. There was almost certainly a significant lag between the appearance of lateen sails and their appearance in this artwork, both because it would take a while for it to enter the iconographic repertoire of artists, and because we don't actually have the oldest depictions, we only have the oldest depictions that we have found. Important distinction! So, based purely on this, we can say that lateen sails showed up in the Mediterranean sometime in the Early Imperial Period.

But there is a somewhat more interesting way of thinking of this. Artists at the time didn't often depict ships with very much concern towards accuracy, nor were they really trying to give ethmographic depictions of ship types. To give a modern illustration, if you type "sailing ship" in to Google you will mainly get images of a big hull leading to a prow, three masts and a bunch of square sails. That is basically the image of "sailing ship" and an artist trying to depict a sailing ship is basically going to be content with drawing something like that. Likewise, a Roman artist trying to depict a general ship will come up with something like this. However, there must have been many different types of ships sailing in the Mediterranean, which can be inferred from modern ethnography as well as literary examples.

So it may very well be that lateen sails had been used for a very long time by a particular seafaring culture, it just didn't happen to enter into the iconographic repertoire until the second/third century.

To add to the excellent response from /u/Tiako, I might point you towards this journal article (PDF warning): http://www.uhpress.hawaii.edu/journals/jwh/jwh061p001.pdf

There are two and possibly three origins of the lateen sail: the Mediterranean and/or the Indian Ocean, and separately Austronesia (later diffused into Polynesia). The PDF I linked covers the argument over whether the lateen was invented in the Indian Ocean and then spread to the Mediterranean, or vice versa, or whether it was independently invented in both places.

The origin of the lateen in southeast Asia and Polynesia seems to have been completely separate, though -- it was an adaptation from a V-shaped "square sail" suspended from two spars, which eventually became attached to a prop, then a canted mast. (The diagrams in the pdf explain this better.)