Well for a while it should. For instance I will use my system as an example...I am using a 8600GT 256MB it also requires 350 Watts. Thing is my system (Dell Inspiron) comes stock with 300 watts. So i took a leap of faith and used the card anyway. It ran fine for a long time. Six months to be exact. Nothing went wrong. The only reason why i said six months was because i bought a new PSU just for few upgraded components. TO answer your question TECHNICALLY it should work.
Only thing i have ever used is geforce, so know nothing about ati or anything.
It looks good to me though, and says 300W requirement.
Review on google said to use this one over the 9400 GT that was posted previously, for gaming.