Numerical simulations of polymer injection molding enable the prediction and mitigation of manufacturing defects in this widely used process. However, these simulations remain challenging, and efficient numerical analysis remains an active research area. A major difficulty arises during the cavity filling stage, when molten polymer displaces air inside the mold. In such two-phase flow simulations, a material discontinuity exists at the moving boundary between the polymer and the air, requiring high spatial and temporal resolution, which leads to considerable computational expense. Following the work in [1]-[3], we propose a mesh refinement strategy in both space and time that localizes the computational effort at the interfacial discontinuity. This approach uses a space-time discretization to achieve localized refinement in both space and time simultaneously. We present this methodology and its application to realistic injection molding simulations. We show how adaptive refinement in time and space can enhance both accuracy and efficiency for this application in particular, and, in general, for interfacial flow problems. [1] V. Karyofylli, M. Frings, S. Elgeti, M. Behr, Simplex space-time meshes in two-phase flow simulations, Numerical Methods in Fluids, Vol. 86, pp. 218-230, 2018 [2] V. Karyofylli, L. Wendling, M. Make, N. Hosters, M. Behr, Simplex space-time meshes in thermally coupled two-phase flow simulations of mold filling, Computers & Fluids, Vol. 192, 2019 [3] B. Ferrer-Fabón, J. Alms, M. Behr, C. Hopmann, High-resolution numerical simulations of polymer injection molding: Analysis of mesh size and refinement, 93th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM), Vol. 23, 2023