Epiphany, as the Webster 1913 definition for this word notes, comes from a Greek phrase meaning, roughly, "appearance, manifestation". However, the context seems to be missing. To a Greek, epiphaneia would imply the literal appearance of a deity to a mortal. The connotation is clearly of religious awe in the presence of the divine.
Epiphaneia is infrequently used in Greek tragedy, most notably by Euripides. In the opening scene of his play Trojan Women, Poseidon and Athena debate punishment for the Greeks for their excesses in the course of conquering Troy. To the Greek audience, this appearance of two deities in the very first moments of the play must have been somewhat shocking - particularly in light of the socially and politically critical theme of the play (which may be read as a rebuke to the Athenians for their behaviour in the Peloponnesian War, particularly with regard to the people of Melos, which was conquered in 416 BCE).