Best answer: It absolutely does. Just as Xbox One controllers have forwards compatibility with the Xbox Series X and Series S, the new controller has backwards compatibility with the Xbox One.
Backwards compatible: New Xbox Wireless Controller ($60 at Microsoft)
Backwards compatible to Xbox One
Microsoft has made a big deal out of compatibility of not only games, but accessories, between the Xbox One, the Xbox Series X and Xbox Series S. And while existing hardware is forwards compatible, the sharing doesn’t end there, because the new Xbox controller is backwards compatible to the Xbox One.
That means if you’re picking up a new controller for your Xbox One, a Series X|S controller is a fine choice, not least because of one of its new features.
Share button functionality works on Xbox One
One of the signature features of the new Xbox Wireless Controller over the previous generation is the hardware share button. And support for this is also backwards compatible to Xbox One, which might be reason enough to pick one up.
On Xbox One with the standard controller, capturing screenshots or clips is a two tap process, first opening the guide then pressing either X or Y depending what you want to capture.
With the new controller, on the Xbox Series X|S or the Xbox One, a single press snaps a screenshot or a long press grabs a video capture. Simple, easy, and no interruption to your gameplay.
Ditch the micro USB cables
Another bonus to the newer controller over the old one is that it embraces USB-C for charging (with a play and charge kit) or wired connection to the console. Since there’s a strong chance you’re now using USB-C in a lot more places than micro USB, you can toss out your remaining old cables.
Get the new controller for your older Xbox
If you haven’t jumped to the Xbox Series X|S yet you can still use the new controller on your Xbox One, complete with full support of the new share button.