Not sure why you think that something tall should fall flat on its face. Crudely put, things fall over when their centre of gravity shifts outside their base. It's difficult to find accurate information about the speed of take-off of the shuttle, but the consensus seems to be that by the time the rocket clear the tower, it's travelling at somewhere between 70 - 140 mph, which is probably somewhat faster than a firework rocket.
It's also a fair bet to assume that the manufacturing tolerance of the vehicle and its engines are a few orders of magnitude tighter than those of a firework. Plus, of course, it doesn't have a stick on one side!