Leveraging coherent light-matter interaction in solids is a promising new direction towards control and functionalization of quantum materials, to potentially realize regimes inaccessible in equilibrium and stabilize new or useful states of matter. We show how driving the strongly spin-orbit coupled proximal Kitaev magnet $alpha$-RuCl$_3$ with circularly-polarized light can give rise to a novel ligand-mediated magneto-electric effect that both photo-induces a large dynamical effective magnetic field and dramatically alters the interplay of competing isotropic and anisotropic exchange interactions. We propose that tailored light pulses can nudge the material towards the elusive Kitaev quantum spin liquid as well as probe competing magnetic instabilities far from equilibrium, and predict that the transient competition of magnetic exchange processes can be readily observed via pump-probe spectroscopy.