The coupling between spin and charge degrees of freedom in a crystal imparts strong optical signatures on scattered electromagnetic waves. This has led to magneto-optical effects with a host of applications, from the sensitive detection of local magnetic order to data storage technologies. Here, we demonstrate a new magneto-optical effect: the tuning of inelastically scattered light through symmetry control in atomically thin chromium triiodide (CrI₃). In monolayers, we found an extraordinarily large magneto-optical Raman effect from an A_1g phonon mode due to ferromagnetic order. The linearly polarized, inelastically scattered light rotates by ~40^o, more than two orders of magnitude larger than the rotation from MOKE under the same experimental conditions. In CrI₃ bilayers, we show that the same A_1g phonon mode becomes Davydov-split into two modes of opposite parity, exhibiting selection rules that depend on inversion symmetry and magnetic order. Applied magnetic and electric fields allow for magnetoelectrical control over these selection rules. Our work highlights the unique opportunities provided by 2D magnets for controlling underlying symmetries to manipulate Raman optical selection rules and for exploring emergent magneto-optical effects and spin-phonon coupled physics.