And since it just now occurred to me before anyone takes out my throat;
Yes, your post-modern Xbox wireless controller works just dandy with your old-school games that didn't know what an xbox was when they were made. That's because controller support is assumed by Windows and Windows can do whatever it wants; including, but not limited to, working properly.
In this day and age every piece of software now has the cahones to out-maneuver Windows and supply it's own support for this, that and the other thing to prevent misuse of the software's design. Of course, I'm speaking of Bottting. Leaving controller configurations and peripherals in Windows' hands allows bot-making and other such malintent to occur more easily and Square (or any other software company for that matter) is forbade to reign in the power of Microsoft's Micromanagement. This is how you wind up with the Siamese Twin Software Syndrome. Two things doing the exact same thing but each is operated by a different BRAIN.