Romans had plenty of evidence that they could clean out a species from a geographical location completely, but it is harder to say that they knew they could make a species extinct as we understand it today. We have to remember that making room for more farms and protecting people against wild animals was seen as a general win.

Ammianus Marcellinus, writing his *Res Gestae* in the late fourth century mentions that hippopotami, once exported from Egypt to Rome, were no longer to be found in the region in Book XXII:

“24. This monster was very rare till it was first exhibited to the Roman people in the aedileship of Scaurus, the father of that Scaurus whom Cicero defended, when he charged the Sardinians to cherish the same opinion as the rest of the world of the authority of that noble family. Since that time, at different periods, many specimens have been brought to Rome, and now they are not to be found in Egypt, having been driven, according to the conjecture of the inhabitants, up to the Blemmyae by being incessantly pursued by the people.” (Taken from an online translation by C.D. Yonge)

The Anthologia Graeca (Volume 2 of W.R Paton’s translation) Book VII, epigram 626 reads “Ye furthest Nasamonian wilds of Libya, no longer, your expanse vexed by the hordes of wild beasts of the continent, shall ye ring in echo, even beyond the sands of the nomads, to the voice of lions roaring in the desert, since Caesar the son has trapped the countless tribe and brought it face to face with his fighters. Now the heights once full of the lairs of prowling beasts are pasturage for the cattle of men” This particular epigram is commonly regarded to have been written no later than the reign of Nero.

A quotation from Themistius (Orationes 10, 140) is given by William Epplett in Animal Spectacula of the Roman Empire is of particular interest:

“And yet in hunting we leave behind their [the animals'] seed, and the one completely removing it is considered to be sinning against the hunt. And so we spare for ourselves the fiercest animals...so that by procreation they might be preserved and remain, and we are angry at the elephants removed from Libya, the lions from Thessaly, the hippopotami from the marshes around the Nile”

He goes on to say that In the fifth century, farmers in the east needed to apply to for permission before killing lions from the emperor.

You could try W.C Epplett’s section on extinction in this paper and J. Donald Hughe’s chapter on Wildlife Depletion in Pan’s Travail - Environmental Problems of the Ancient Greeks and Romans for further reading.