If there are such works, how much do we know about them? I would also be interested in hearing about, say, plays or poems that are popular today because they're the only surviving ones we have from certain cultures.
Well, sort of. The Epic of Gilgamesh had enormous importance in the ancient world, and was referenced in many early documents. But we did not have any actual account of Gilgamesh beyond a few fragments. That went on for awhile, but then in the mid-19th century, a longer portion was found and translated. Since that time, even more has been found, though it's still not complete.
You might find this page about the survival of sources from antiquity really helpful. If you want to fall down a rabbit hole, the Popular Questions section is littered with interesting information and discussions.