We present the first example of magnetic ordering-induced multiferroic behavior in a metal-organic framework magnet. This compound is [CH3NH3][Co(HCOO)3] with a perovskite-like structure. The A-site [CH3NH3](+) cation strongly distorts the framework, allowing anisotropic magnetic and electric behavior and coupling between them to occur. This material is a spin canted antiferromagnet below 15.9 K with a weak ferromagnetic component attributable to Dzyaloshinskii-Moriya (DM) interactions and experiences a discontinuous hysteretic magnetic-field-induced switching along [010] and a more continuous hysteresis along [101]. Coupling between the magnetic and electric order is resolved when the field is applied along this [101]: a spin rearrangement occurs at a critical magnetic field in the ac plane that induces a change in the electric polarization along [101] and [10-1]. The electric polarization exhibits an unusual memory effect, as it remembers the direction of the previous two magnetic-field pulses applied. The data are consistent with an inverse-DM mechanism for multiferroic behavior.