I don't see why it wouldn't work. I'm running my PS3 in HDMI -> DVI and it looks great. It should work the same for the Xbox 360.
I also used an Xbox 360 composite -> VGA and that worked fine as well (in case you want to try VGA instead of DVI). Purists will tell you that the image quality of VGA is worse than DVI and DVI isn't as good as HDMI; well, the truth is you probably won't be able to tell the difference. It's all 1080p and still looks great.