Due to its technological importance, controlling the perpendicular magnetic anisotropy (PMA) in thin films has received considerable attention in recent years. PMA based devices usually involve heavy-metal (oxide)/ferromagnetic-metal bilayers, where, thanks to interfacial spin-orbit coupling (SOC), the in-plane (IP) stabilization of the magnetization is broken. Here we show experimentally that in V/MgO/Fe(001) epitaxial junctions with competing in-plane and out-of-plane (OOP) magnetic anisotropies, the SOC mediated interaction between a ferromagnet (FM) and a superconductor (SC) enhances the effective PMA below the superconducting transition. For the larger area junctions with more stable IP magnetization, the OOP anisotropy field can be increased either by the interaction between magnetic stray fields and superconducting vortices or decreased by an applied electric field. Our findings, supported by theoretical and numerical modelling of the ferromagnet-superconductor interaction, open pathways to active control of magnetic anisotropy in the emerging dissipation-free superconducting electronics involving spin.