Aristophanes was a playwright, poet and reformer in fifth century BC athens, and, while he didn't "invent" burlesque itself, he set in motion the attitudes and ingredients that would later become the skeleton of the art.
He was famous for his powers of ridicule, comic, puns and satire (the word 'burlesque' actually comes from the word 'burlare', which means 'to laugh at, to make fun of'). In his plays, he made fun of real-life enemies and, although all actors in classical Athens at that time wore masks when they performed, Aristophane's masks were not of stereotypical characters but of real people, thus all the easier to make fun of due to the fact that they could be easily identified.
Just as in burlesque today, the audience was often drawn into the action, whether with seductive involvement or mockery. Dirty jokes, teasing and music were also implimented in his plays.
But the one play which many, including the current Queen of Burlesque, Dita Von Teese, believe to be the one to watch, is called 'Lysistrata' which tells of the wives of Athenian soldiers hiding away in the Acropolis, refusing their husbands sex until the execution of the Peloponnesian War. The play is notable for its exposé of sexual relations in a male-dominated society and for its use of explicit obscenities.
Not only is this a fine example of female power in ancient civilisation but, as Dita mentions in her book, these woman are teasing their husbands, guiding their minds towards sex and then locking it away, just as a good burlesque performer does.
It may be a primitive form of burlesque but, just as Calvinism gave societies the attitudes needed for Capitalism, Aristophanes gave the world the attitude it needed for the birth of burlesque.
Burlesque, The Art of the Teese - Dita Von Teese