Use a power supply as a switch
The simplest solution to this is a relay. Say the existing fan is 12V d.c. (as per PCs), wire a relay coil across the existing fan connections, in parallel with it, and wire the (normally open) contact pair in series with your power supply and the new fans eg break the +12V line of your new power supply. It is safer for you to do this by breaking the 12V line to the fans than breaking into the mains live terminal to your power supply since it reduces your risk of electrocution and possible damage to your Xbox if a wire comes loose, but it means the new supply is running all the time.