In the Greco-Roman world, Thales of Miletos is widely credited with being the first person to identify lunar eclipses as being caused by the earth's shadow. This was in the early 6th century BCE. The nature of his argumentation and/or evidence for the theory is lost, unfortunately.

It seems that his theory wasn't universally accepted for a while -- within the 6th century there are competing (and wrong) interpretations of the nature of the moon from Anaximander and Herakleitos -- but once knowledge of the earth's sphericity became widespread in the 4th century, it was seen in hindsight that Thales' interpretation was the only possible one.

This theory certainly postdates the prediction of lunar eclipses, which is pretty trivial and doesn't require any particular model of the spatial relationship between sun, earth, and moon. Prediction was well within the capabilities of Babylonian astronomy before Thales came along. (Solar eclipses, of course, are another matter -- but there are some indications that even they might have been at least semi-predictable for Babylonian astronomers of the 5th century BCE and later.)