Yes it will still run but you'll be pushing your PSU to its limits.
But since you have only 400W PSU I would strongly recommend you a 5770 over a 260 because of the lower power consumption. 5770 will also dissipate lesser heat . And with more mature drivers it will perform better.
The 260 is about 10% faster than the 5770 on average in DX9/10 games and both fare well at resolutions up to 1920 x 1200. The 260 does not support DX11 ... the 5770 can do Dx11 up to about 1650 x 1050 resolution
The 260 pulls 119 watts
See if your pus can handle it by using this calculator to 'do the math" for you