The family of generalized Paley graphs of prime power order $q$ and degree $(q-1)/k$ is studied. It is shown that the automorphism group of a graph in this family is a subgroup of ${\mathrm{AΓL}}(1,q)$ whenever $q$ is sufficiently large relative to $k$. Furthermore, under the same conditions, the Weisfeiler-Leman dimension of these graphs is proved to be at most $5$. In particular, the same bound holds for the Van Lint-Schrijver graphs.