It will, this is a great PSU. In fact, 770 can work on 600W PSU without any issues, provided that the PSU is a quality one. Buying a more powerful PSU is smart thing, though, your PSU won't be loaded so much as 600W.
Awesome, I decided to go with the 750W instead of the 600W just so I'm not bottlenecked later if I decide to SLI or something later on. I was just worried because I had a 600W PSU, but it didn't have a 12V rail for the PCIe and it wouldn't work with the 770. AND my old PSU only had a 6-pin PCIe power cable and only one 6+2 pin cable, while the 770 needed two 8-pin connectors.