The backwards compatibility of the Xbox One is one of the coolest features about the system.
On the other hand, there is no forward compatibility, so to speak, when it comes to the Xbox 360 hardware.
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Xbox 360 controllers do not work on an Xbox One, so you'll need an Xbox One controller
You cannot use an Xbox 360 controller to play games on an Xbox One.
Other accessories from the older system are likewise incompatible with the Xbox One.
So if your Xbox One controller breaks, don't try digging out the old system to use an old controller. You'll have to buy a new (or used) one.
How to get a new Xbox One controller inexpensively, or find a used one
Yet another option is to consider auction sites like eBay or a retail store like GameStop where great deals are often available on used but perfectly serviceable Xbox One controllers.