Ian Morris tries to answer this question in "Why the West Rules for Now" as well as in some earlier work on the ancient Mediterranean. The basic argument says that the earliest civilizations were structured civilizations. i.e. there was a central economic authority which told everyone what to make and then redistributed resources. This type of system is evident to monarchies where the King could do what he wished with all economic resources. He gives examples of this with the Minoan and Mycenean civilizations.

Then, after the Greek Dark ages, a new system appeared. The market system. The argument is that the market system is more efficient than the centralized system and the Greek world achieved economic growth through its market institutions rather than through technological progress or population increase (the major drivers of today's economies). The market system and democracy go hand in hand, democracy being a market of political power, so the political system evolved with the economic system. In addition, to make this market system go smoothly, coinage was invented (supposedly in ancient Lydia) which made markets easier and more efficient, adding to their advantage. According to Herodotus, the Persians in fact considered the Greek market system backwards and he says that they had no markets of their own (doubtful, but the fact that he would even remark at something like this gives us a hint that the Greek knew that the market system was something important).

The efficiency contributed to the dominance of the system which has survived all the way down to today.