I have an Xbox 360 and yes my monitor is connected to my video card.
Here Is a picture of the back of my monitor. The HDMI to DVI is the output coming from my Roxio. VGA is connected to Video Card.
Here is the Roxio. The output to my monitor, the input from my xbox, and USB connected to my rear IO.
Showing that the HDMI is truly hooked up to the back of my XBOX.