eh
I would not try to put that card in with a 375 PSU. Of course it also depends on what processor you have and fans. However, even with a low wattage processor, you could still run into an issue if you run your computer full load and all the components are sucking that power up. I would recommend a new power supply, about 50.00 US dollars for 450W. You might be able to even get a green 80+ power supply.
To summarize, I would not run a 4770 with only 375W power supply.