Yeah I don't see a reason why it wouldn't work, and you will get more GPU power from the 1060, BUT, in many games your cpu WILL bottleneck the GPU (speaking from experience, I have FX-8350 w/ 8gb 1600Mhz ram) and in conclusion, you won't get that much FPS boost in games as expected.
If you're doing just 3d rendering etc. where only the gpu is under load, I don't see a reason why not to upgrade, but if it's only for gaming, I would look in to new cpu, even older intel cpu's will have a better performance in games and will not bottleneck the GPU, atleast not nearly as much as pretty much all the FX-series cpu's will.