Yes, a black Xbox 360 controller works on the Xbox 360 Premium Console—provided it's a genuine Xbox 360 controller.
The "Premium Console" is just a 2005–2007 Xbox 360 SKU (20GB/60GB HDD, chrome tray). It uses the same controller protocol as all 360s.
Bottom line: If it's a real Xbox 360 controller (black, white, pink, Halo edition, etc.), it works. If it's an Xbox One controller, it doesn't.