Employing an exact diagonalization Monte Carlo solution of the first-principles-derived model Hamiltonian of a number of A2BB′O6 double-perovskite compounds, containing a 3d transition metal at the B site and a 4d or 5d transition metal ion at the B′ site, we investigate the effect of B/B′ cation disorder on their magnetic properties. Our exhaustive study reveals that the influence of cation disorder on both the magnetic transition temperature and magnetization depends strongly on the underlying exchange mechanism with a distinct difference between the double exchange mechanism and a combined double exchange and superexchange mechanism. We further find that the nature of the disorder has a remarkable effect. While the uncorrelated or random disorder has a severely detrimental effect, especially for magnetism having a superexchange contribution, correlated disorder with a high degree of short-range order retains the magnetic properties of the fully cation-ordered compounds to a large extent. Our findings shed light on the puzzling report of magnetic order in fully cation-disordered CrRu oxides.