PC doesn't start with graphics card

[Feali]

Okay this might sound stupid and it's probably just the bad power supply. But maybe someone can help me with this:
So I have this Alienware X51 R2 (yes i know; biggest purchase mistake) and everything was going fine for 1 1/2 years but today something very odd happened. After I came back from lunch my PC had apparently turned itself off. Out of nowhere. I was just sitting in the Heroes of the Storm Hero Selection screen so nothing too fancy. I of course immediately tried to turn it back on just to find out that this didn't work.
I unplugged the power chord, waited several minutes for everything to cool down. Still nothing. So I opened the PC. Nothing was wrong as I far as I could see but I'm no expert.
After closing it up I tried to turn it on again and it still wouldn't work. Since I have an onboard graphics card I just took the GTX 670 out and started the PC again. Lo and behold it worked! So I thought well maybe the problem is fixed but everytime I try to start the PC with the 670 it won't work.

Some specifics to help you (mind: someone already told me 330W for that setup is way too low but it was the only choice for this piece of crap):
Intel Core™ i7-4770 (8MB Cache, 3.9GHz)
NVIDIA GeForce GTX 670 2 GB GDDR5
16GB Dual Channel DDR3 at 1600Mhz
2TB (64MB Cache) 7200 RPM SATA 3Gb/s
330W external AC/DC Adapter (there is no external 500W; and building on in isn't a choice)

[skitszo]

if you have a second computer swap out power supplies; or order a PSU tester from amazon.

take the card and try it in a different rig with a better power supply....

get a new graphics card or a different one you have access to and try installing it to see if it works....

you can go thru the motions of reseating everything; pulling out memory and reinstalling; unattaching power cables and reinstalling etc etc. blow the thing out.

from you description i'd go with either the graphics card or PSU....motherboard becomes suspect when you rule everything else out as being OK.

- - - Updated - - -

price picker estimates your power needs to be 352 watts..... 20% +20% = you should have a 80@ bronze PSU of 490 watts...so 500 watt psu.

thats if my math works right...roughly....

so i'd say your PSU might need to be replaced...hopefully you didn't burn out something on the motherboard or the graphics card.