If it seems hard or even impossible to defend the American way on the merits, that's probably because it emerged from aesthetic, not logical, considerations. According to Rosemary Feal, executive director of the MLA, it was instituted in the early days of the Republic in order "to improve the appearance of the text. A comma or period that follows a closing quotation mark appears to hang off by itself and creates a gap in the line (since the space over the mark combines with the following word space)." I don't doubt Feal, but the appearance argument doesn't carry much heft today; more to the point is that we are simply accustomed to the style.
It is not proper practice in American grammar to put the period outside of quotation marks, but that is the way the Brits do it. Should we change? Ben Yagoda says yes, we should (via Gruber):