By default, no. Most games only look for controllers that use the XInput API, or essentially, the x360 controller layout. Typically this can be gotten around by using a wrapper that maps the PS4 (or any other joysticks) input to match that of the 360, essentially tricking the system to think you have a x360 controller plugged in.
Granted, it's better then it was even a year or two ago, but I wouldn't expect any joypad not made by MSFT to universally work in games without at least some external tools.