Using first-principles calculations based on the density functional theory, we show a strong strain dependence of magnetic order in (LaMnO3)n/(SrMnO3)2n (001) superlattices with n=1,2. The epitaxial strain lifts the degeneracy of Mn eg orbitals, thus inducing an inherent orbital order, which in turn strongly affects the ferromagnetic double exchange of itinerant eg electrons, competing with the antiferromagnetic superexchange of localized t2g electrons. For the case of tensile strain induced by SrTiO3 (001) substrate, we find that the ground state is A-type antiferromagnetic and dx2−y2 orbital ordered, which is in excellent agreement with recent experiments [ May et al. Nat. Mater. 8 892 (2009)]. Instead, for the case of compressive strain induced by LaAlO3 (001) substrate, we predict that the ground state is C-type antiferromagnetic and d3z2−r2 orbital ordered.