First of all, DON'T BUY the ridiculously overpriced original Microsoft cable, the cheap clones are usually just as good and they are like one tenth the price.
In the beginning I used such a cable for my 360 as well and it works okay. You will probably need an additional Cinch stereo jack adapter though if you don't have one already.
But you will not get the best possible picture quality when you're using that VGA cable. Since your monitor has a DVI port, you would get the best picture quality by connecting the 360 HDMI output to the DVI input of your monitor.







