Oxygen octahedral rotations have been measured in short-period (LaNiO$_3$)$_n$/(SrMnO$_3$)$_m$ superlattices using synchrotron diffraction. The in-plane and out-of-plane bond angles and lengths are found to systematically vary with superlattice composition. Rotations are suppressed in structures with $m>n$, producing a nearly cubic form of LaNiO$_3$. Large rotations are present in structures with $m<n$, leading to reduced bond angles in SrMnO$_3$. The metal-oxygen-metal bond lengths decrease as rotations are reduced, in contrast to behavior previously observed in strained, single layer films. This result demonstrates that superlattice structures can be used to stabilize non-equilibrium octahedral behavior in a manner distinct from epitaxial strain, providing a novel means to engineer the electronic and ferroic properties of oxide heterostructures.