I am having the same issue after the newest patch. My Xbox 360 controller worked prior to the patch and now it does not. I do not have any other input devices (except for a mouse and keyboard) plugged in. I spent an hour talking to a support rep and they were no help. Apparently they filed a report with a specialist team to invesitigate the issue, but I'm doubting any sort of resolution any time soon. Such a sad thing to happen so soon after the release of this game, especially since it was something that was working prior to the patch.
It would be nice to see an input device selection in the options menu.
For a quick fix, edit the settings to use the mouse and keyboard for the time being. If you are stuck and can't get to the options menu, navigate to Documents/bioware/dragon age inquisition/save/profileoptions_profile and change the value associated with the setting with "gamepad" in the name to 0 (should be 1 if your gamepad setting is enabled).