It’s possible you might be able to get it to run, but far from guaranteed, as Intel’s OpenGL support is apparently fairly flakey.
If you *did* manage to get it running… the best I’ve heard about the experience is “playable”… and I get the feeling that’s at the “barely” end of the spectrum.
Basically, you need a GeForce or a Radeon for any reasonable gaming.
That 350w power supply might be a problem, depending on what brand it is. If it’s a reasonable named brand, then maybe try picking up one of the new nVidia Maxwell-based cards – GeForce GTX 750 or GeForce GTX 750 Ti – which have power requirements of 55 and 60W respectively, and so won’t even require additional power connectors. They’ll also be fairly cheap, and will run most games at decent settings.