Inclusion-exclusion can be thought of, in some sense, as a special case of the use of generating functions, but both should be understood in a fairly broad sense for this to be true. A broad context for inclusion-exclusion is Möbius inversion on a poset, which reduces to ordinary inclusion-exclusion in many cases such as when the poset is the poset of subsets of a set. Möbius inversion naturally takes place in the incidence algebra of a poset, which can be thought of as a place where generating functions associated to the poset naturally live. In particular, three of the most important types of generating functions (ordinary, exponential, Dirichlet) can be thought of as associated to posets in roughly this manner, although the details are a bit subtle.
For details, see Doubilet, Rota, and Stanley's On the foundations of combinatorial theory (VI): the idea of generating function.