The external power cable (PSU outer side to wall) is usually interchangeable for personal computers. PSUs are generally sold with a fresh power cable with a NEMA 5-15P plug, but it's usually fine to use the old one. Some cables have a maximum power rating printed on the outside.
The internal cabling (PSU inner side to motherboard or other components) is not guaranteed to be interchangeable. A PSU is either modular or non-modular. Modular means that there are plugs on the inner side of the PSU, and those will vary by model. Non-modular or cabled means that the cables come straight out of the inner side of the PSU, so those cannot be removed. http://www.buildcomputers.net/modular-psu.html
It's good to ask on tech sites like SuperUser, but the vendor is the most reliable resource for sure answers. I suggest going to Corsair with this question, and adding your own Answer below afterward.