US basically dictates global policies in the block of countries not part of the soviet sphere of influence during the cold war. US defends it's position in the global community using every trick and leverage available. If something is illegal in the US you can be quite assured they will try to apply leverage to make it illegal everywhere else, especially if it's an issue that requires international co-operation - like the war on drugs, for example.
For example, the international banking system is basically controlled by US policies. Parties, that do not co-operate and obey US authorities can and will be shut out of the large pool of global financial services that is critical to running businesses effectively.
Another effect is the pure cultural force US practices have on the "western block" of countries. Lots of ideas (good or bad) are taken into use in other countries after they become fashionable in the US.
You are quite right prudish moral superiority is of course a facet, but that is not the key here, I think.