You are thinking of the still fully extant copies. For the Illiad we have one that was made in tenth century Venice. However we also do have damaged excerpts surviving from the Hellenistic age, discovered more recently, which can be used to confirm (or call into question) the copies that we already have.

Aside from that, the Grecian authors are cross referencing one another. We have, for instance, secondary sources giving a full list of authentic platonic dialogues which were used (along with linguistic evidence from inside of the text) to sort out the dialogues that were held for Platonic for a very long time.

From the quotations from Homer in Aristotle and Plato we can tell our copy is different. One way you can explain is they were quoting inexactly from the memory. Another is that our copy had not been canonical yet in their time.

Hi, you might be interested in this bit I wrote on textual criticism.

Note that unless the text in question itself names the author, text crit can't help with determining authorship on its own. Once we have a good text analysis of style and stuff can help us judge whether a traditional attribution is good.

As for the text itself, even if all our manuscript copies are very late (and some, for instance, Cicero's de Re Publica, exist in only one manuscript), the indirect tradition (that is, quotes preserved in other sources) and papyri confirm the texts are ancient and many of our reconstructions correct.