I have a 5770 running off of a 380 watt, and the 7750 uses a good 30-40 watts less. As well as your CPU is probably using less than mine too. I would check your power supply stats and the sticker on the side and ensure that the +12v amp rating is in the 20-24 amp range and at that point I think you would be fine. I actually do have a power meter coming in the mail hopefully being delivered today since I'm thinking of throwing a 6870 on to my 380 and I want to be sure it isn't too much. People shouldn't recommend what AMD says as the minimum recommended, because they have to say far more than what's actually needed to cover for garbage power supplies that melt when you push 60% of their ratings through them.
I had borrowed a kill-a-watt a while back and found my i5-3570k, 8GB, 7850, 2x7200 and 1xSSD peaked at using 230 (160-175 while gaming) watts from the wall. At 85% efficient that means my PSU is actually delivering about 195 watts, and even then if the entire amount was coming off of the 12v rail it would only peak at 16.25 amps.