Do any complete manuscripts of Ancient Greek plays survive?

Yes - as the above answers make clear. Yet what it's also worth making clear is that there's little logic behind why these particular plays survive, and that the plays that do survive are only a tiny percentage of each playwright's actual output. We also know of several other playwrights writing at the time - but for some reason their work has not survived.

We know that Sophocles wrote over 120 plays of which only the seven listed above survive in full. We also, however, have fragmentary versions of several others: including from his "Hermione", "Troilus" and "Niobe".

We have eighteen of the ninety-five plays Euripides penned, included the only complete surviving satyr play, "Cyclops" (the satyr play was the comedic play which followed the performance of three tragic plays in Greek dramatic festivals).

And we have only seven of the ninety-two that Aeschylus penned: including the "Oresteia" ("Agamemnon", "The Libation Bearers", and "The Eumenides") which is the only surviving complete trilogy (three plays meant to be performed together).