You could rename the macro 'FileSave' (and make a copy that's named 'FileSaveAs' - plus adding the necessary code to deploy the 'Save As' dialogue) to intercept Word's FileSave & FileSaveAs events, but more recent Word versions seem to be deprecating those. However, since the Document_Open event achieves much the same result for any document upon opening, I'd have thought that would be sufficient - the only thing it won't automatically update is the filename when a file is first created or re-saved via Save As, but it will capture that too, the next time the document is opened.
Although Word has 'DocumentBeforeSave' and 'DocumentBeforeClose' application events you could use, programming those is rather more complicated than using the built-in Document_Open event. To see what's involved, check out:
http://word.mvps.org/FAQs/MacrosVBA/AppClassEvents.htm