Yes, you can. All Xbox 360 accessories (av cable, ethernet cable, memory unit, controllers, etc.) and games work with all versions of the xbox 360. The only exclusion I know is the HDMI cable which only works with Xbox 360s with built in HDMI port (all xbox 360s versions currently being sold have HDMI ports).
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The same way you connect it to a regular Xbox 360.
One is Black, the normal wireless controller is White.
xbox 360 elite
The Xbox 360 controller will not work on an original xbox console. Similarly, an original xbox controller will not work with the xbox 360. Each controller only works with its respective console. Xbox 360 Controller --> Xbox 360 Xbox controller --> Xbox
You can sync your controller into your xbox 360.To do that there are certain buttons on the controller and the xbox 360 you push to sync the controller.