It isn't a hardware fault but a driver problem. Intel drivers seem to do this sort of thing quite often, there have been a few reports of issues.
Connect both the DVI and analogue and boot into Windows, so you will see the screen using the analogue connection.
Right click on the desktop and chose screen resolution. Do you see two monitors here? If yes click the Identify button, as you are seeing the desktop by the analogue connection you now know which one it is, either 1 or 2.
Under the multiple displays drop down, set it to "Show desktop only on x" where X is the opposite of the analogue display. Click Apply. Your screen should flicker, you might need to manually switch your monitor back to DVI now to see if the display is working over DVI again. If not, switch back to the analogue input on your monitor and after 15 seconds Windows will set back to analogue again so at least you are at the desktop and not left high and dry.
Hopefully that will help.
I have a very similar but much worse problem with my 3570K. I want to use only the internal HD 4000 graphics. I started out with a Gigabyte GA-Z77X-UD5H motherboard and could load Win7x64 and everything would work just fine if I used the analog VGA port. But if I ever loaded the Intel video driver (tried all versions) the screen would go black and I could never get any of the digital ports to work at all. You could move the connector from the VGA port to DVI while the screen was on and nothing. I assumed I had a bad motherboard.
I then got an ASUS P8Z77-V Pro and it wont even power on with this same CPU unless I install a PCIe graphics card. I can't seem to get the internal graphics ports to work on it either. So at this point I think there must be something wrong with the HD4000 graphics on my CPU. I am thinking of RMA'ing the processor next.
The Gigabyte board would at least boot but maybe there are some more detailed checks in the ASUS BIOS. It doesn't even beep. But if you plug in a PCIe card it fires right up so the main CPU and memory is fine.
I have a 3770K and neither my graphics card nor the HD 4000 works for me (any version I try, it will not boot past windows screen). When i use my external graphics card, it will SOMETIMES boot past the log-in, but is very unreliable. I first used an ASUS P8 Z77-V LK mobo, replaced my PSU, got a new graphics card...and now I am using a Gigabyte Z77X-UD3H mobo with the same problems. With all of this hardware replaced, I still have the same problems. I'm thinking I should have gotten an older more reliable processor? It's the only thing that's defective that I can think of, it's a really weird problem.
I've been troubleshooting this for weeks and the post above is the closest to my problem that I've seen.
Basically, like gff, I'm stuck using the default Display Adapter VGA, display is very poor can't play any games naturally.