I can only speak for Ancient Greece - but there, at least, I can confirm that applause (κρότος χειρῶν) was common by the late fifth century BCE.

For example, in Aristophanes' Frogs (which premiered in 405 BCE), we have this mention of applause (156-7):

...and happy choirs of men and woman who loudly clap their hands (κρότον χειρῶν πολύν)

Likewise, in Xenophon's Anabasis (written about 370 BCE, but recording events about thirty years earlier), we hear that a dancing girl's performance was applauded (6.1.13). Theophrastus characterizes "the aggravating man" as one who "in a theater, will applaud when others cease" (Characters, 19.10).

At a cursory search, I couldn't find any older references, but it's quite possible I'm overlooking something.

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