A fantasy story is one that has at least one element that is clearly and unambiguously magical or supernatural in nature. This is the definition that I use and, by and large, it includes most stories that are classified as "horror".
Before going any further, I am going to put up a special case, some might argue it to be an exception, of/to that rule. There are stories written as "science fiction" which contain elements of what I call "higher order reality". Things such as psionics and pure energy beings could be classified as fantastical elements, but if the author sticks to a pure science fiction story line, I just call them "elements of some higher order or aspect of reality that cannot be directly viewed or measured using nothing but Newtonian/Einsteinian physics". Thus Zenna Henderson's The People stories, the Q continuum, quite a few elements from Babylon 5, and of course, The Force, can all be squeezed into the science fiction genre.