Using first-principles calculations based on the density functional theory, we show a strong strain dependence of magnetic order in (LaMnO$_{3}$)$_n$/(SrMnO$_{3}$)$_{2n}$ (001) superlattices with $n=1,2$. The epitaxial strain lifts the degeneracy of Mn $e_{g}$ orbitals, thus inducing an inherent orbital order, which in turn strongly affects the ferromagnetic double exchange of itinerant $e_{g}$ electrons, competing with the antiferromagnetic superexchange of localized $t_{2g}$ electrons. For the case of tensile strain induced by SrTiO$_3$ (001) substrate, we find that the ground state is A-type antiferromagnetic and $d_{x^2-y^2}$ orbital ordered, which is in excellent agreement with recent experiments [S. J. May {\it et al.}, Nature Materials {\bf 8}, 892 (2009)]. Instead, for the case of compressive strain induced by LaAlO$_3$ (001) substrate, we predict that the ground state is C-type antiferromagnetic and $d_{3z^2-r^2}$ orbital ordered. ; Comment: The paper is accepted for publication in Phys. Rev. B