Randomly end-linked copolymer networks (RECNs) provide a robust route to self-assembled cocontinuous nanostructures. Here, we study the orientation of cocontinuous polystyrene/poly(d,l-lactide) (PS/PLA) RECNs induced by uniaxial stretching above the glass transition temperatures of the components. Small-angle X-ray scattering (SAXS) reveals that the domains initially undergo nonaffine stretching at low strain (ε < 0.4), followed by domain rotation at larger strains, yielding a "soft elastic" response and providing a high degree of orientation. Transmission electron microscopy (TEM) tomography confirms that stretching leads to topological changes in the nanostructure, corresponding to reorganization of domain interfaces. The combination of orientation at the molecular and nanostructural levels provides substantial improvements in yield strength, toughness, and stiffness. In addition to possibilities for improving mechanical properties, cocontinuous nanostructures with controlled levels of orientation have potential in a variety of contexts including directional ion transport and energy absorption.