That's weird, I had to switch out my X-Box power cable over the summer (after I finally got around to request the replacement cable) and I haven't had any problems like that so far. I would imagine that you just received a defective power cable and that Microsoft would probably send you a replacement one for free. Their customer service (at least for X-Box) has been pretty good whenever I've had to deal with them, so I would wait to see what they say.
If they don't get back to you, you might be able to buy a replacement one at EB Games or something - they might carry the "new" power cords.