We construct reduced order models for two classes of globally coupled multi-component oscillatory systems, selected as prototype models that exhibit synchronization. These are the Kuramoto model, considered both in its original formulation and with a suitable change of coordinates, and a model for the circadian clock. The systems of interest possess strong reduction properties, as their dynamics can be efficiently described with a low-dimensional set of coordinates. Specifically, the solution and selected quantities of interest are well approximated at the reduced level, and the reduced models recover the expected transition to synchronized states as the coupling strengths vary. Assuming that the interactions depend only on the averages of the system variables, the surrogate models ensure a significant computational speedup for large systems.