If you have the quest marked as "completed" in your log, you have the win. Otherwise, the quest is marked as Current.
The quests are: A Goldstruck Gigas, To Paste a Peiste, Megadrile Menace, A Fluttery Fiend, The Beast of Bastore, A Delectable Demon, A Beaked Blusterer, A Man-eating Mite, or An Ulcerous Uragnite.
Note - you don't get the quest marked as completed UNTIL you have exited the zone via the maw. So while I've beaten Glavoid, I still don't have the Tahrongi clear officially because I keep forgetting to exit via the maw