Surviving Greek literature covers a wide ground, and while it doesn't always conform to contemporary genres, there are definitely some horror elements in surviving Greek works. S. T. Joshi in the first volume of Unutterable Horror: A History of Supernatural Fiction points to the manifold monsters of The Odyssey and the stories of Jason and the Argonauts, and the Theogeny of Hesiod as examples of how supernatural horror became a part of Greek literature and culture. Even here, the horror is more often mixed with sorrow, such as when Odysseus encounters his deceased mother and attempts to hug her, but cannot:

She said; I, ardent wish’d to clasp the shade

Of my departed mother; thrice I sprang

Toward her, by desire impetuous urged,

And thrice she flitted from between my arms,

Light as a passing shadow or a dream.

Then, pierced by keener grief, in accents wing’d

With filial earnestness I thus replied.

• The Odyssey, Book XI

Ghosts or shades of the living were present in Greek literature since at least Aeschylus Persae (c. 472 BCE) which featured the shade of Darius, and were fairly explicitly as they had been in life when they died, but could sometimes by necromancy be communicated with or answer questions. For a comparative look at the Greek underworld and how it differed from contemporary views, I would recommend Alice K. Turner's A History of Hell - a bit dated, but still a great survey.

Physical horror is harder to justify, but there were certainly terrible moments that might have shocked or chilled the blood. In Euripides' Medea, when Creusa is poisoned, and an old handmaid reacts:

Among the handmaids was a woman old

And grey, who deemed, I think, that Pan had hold

Upon her, or some spirit, and raised a keen

Awakening shout; till through her lips was seen

A white foam crawling, and her eyeballs back

Twisted, and all her face dead pale for lack

Of life: and while that old dame called, the cry

Turned strangely to its opposite, to die

Sobbing.

Roman literature translated and adapted a good deal of ancient Greek lit., and what we have surviving includes several works which directly mirror or borrow from several of the elements of supernatural horror in the Greek corpus. For example, Virgil's Aeneid borrows the descent to the underworld from the Odyssey, L. Annaeus Seneca wrote treatments of Euripedes' Medea and other plays, and it's worth noting that these works in turn - translated into more contemporary European languages - were a great inspiration to Early Modern writers such as John Webster, author of The White Devil (1612) and The Duchess of Malfi (1614). So the ancient tradition, through transmission, continues to influence us today.