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375W sounds like it might be coming out of a Dell XPS unit . If it is, the Dell XPS PSUs are typically rock solid and are actually slightly under-rated for their output, so a 5770 should be fine.
We also need to know you other system specs to factor out how much power is going to be used by the CPU mobo etc. TBH if it was me I wouldn't put a 5770 on a 375W.
Let us know how it turns out. My mate has an old Q6600 420 sitting around and I've been trying to get him to upgrade the gpu but he insists the psu isn't good enough. Worst comes to worst, you should switch out the RAID0 set up for an SSD + a low rpm hard drive. Should save you a reasonable number of watts from the stock Deskstars they use in Dells.