I would suggest always making sure you have the controller plugged in before launching the game for the first time, as well as making sure you make absolutely no keyboard/mouse input after launch. Your computer is sending the game 360 controller input no different than a 360 controller would, if it doesn't respond then either you haven't configured it properly or the game isn't properly checking for controllers and you'll have to troubleshoot with the game's settings.
If you have a controller that can emulate a 360 controller (Dualshock 3 and 4 included), then a game that supports a 360 controller will ALWAYS be playable via that method. The game's just not made well enough to properly check for controller input.
