In the existent study, combined magneto-convection heat exchange in a driven enclosure having vertical fin was analyzed numerically. The finite element system-based GWR procedure was utilized to determine the flow model’s governing equations. A parametric inquiry was executed to review the influence of Richardson and Hartmann numbers on flow shape and heat removal features inside a frame. The problem’s resulting numerical outcomes were demonstrated graphically in terms of isotherms, streamlines, velocity sketches, local Nusselt number, global Nusselt number, and global fluid temperature. It was found that the varying lengths of the fin surface have a substantial impact on flow building and heat line sketch. Further, it was also noticed that a relatively fin length is needed to increase the heat exchange rate on the right cool wall at a high Richardson number. The fin can significantly enhance heat removal performance rate from an enclosure to adjacent fluid.