The top comment in this AskHistorians thread has the same answer I started with: the word 'applause' comes from the Latin plaudire, to strike/applaud/approve, and Plautus, a Roman playwright of the 2nd century BC, often had his characters call for applause (nunc, spectatores, valete et nobis clare plaudite - now, watchers, fare well and applaud us clearly) at the end of his plays.

That's not the end of it, though: the Roman dramatic tradition built on the Greek, and Greek audiences also applauded; if you follow the askhistorians link further, you'll find a discussion of the Greek words indicating audience approval, and a quote from Plato's Laws, written around 348 BC, complains that audiences used to be dignified, without clapping:

Our music was once divided into its proper forms...It was not permitted to exchange the melodic styles of these established forms and others. Knowledge and informed judgment penalized disobedience. There were no whistles, unmusical mob-noises, or clapping for applause.

But we can go back further. This article, which is a neat history all on its own, references the Old Testament - specifically, 2 Kings 11:12 - in which the coronation of King Jehoash (circa 837 BC) is applauded:

And he brought forth the king's son, and put the crown upon him, and gave him the testimony; and they made him king, and anointed him; and they clapped their hands, and said, God save the king.

That might not be the sort of 'performance' you had in mind, but the Books of Kings were written in the 7th or 6th centuries BC.