Will this PSU properly support my system's power needs?

I agree. You want a high quality PSU, and that certainly isn't one. Get any 500W+ unit from Corsair, Antec, SeaSonic, Silverstone or a 600W+ from OCZ. I say a larger unit from OCZ because they're very cheap, but don't have the highest quality in the lower wattage ranges.

Also, you may want to consider getting a larger unit (like a 650W one) to keep your options open in terms of upgrades. Also, a larger, high quality unit could easily be transferred to a new build.