Saw a ton of people having issues trying to play with non-xbox controllers (I was having too), and just discovered how to without 3rd party tools. All you have to do is enter Big Picture mode, and start the game from inside it. Kinda sucks, but it works.
It works because the game only supports XInput, which is an old API, instead of the new DirectInput, which is what those controllers use. Big Picture, in order to support older games, ensure your controllers always use older API.
You may have to go to 'Settings -> Controller' to enable support for your specific controller.