In this paper, we construct a two-dimensional third-order space-time conservation element and solution element (CESE) method and apply it to the magnetohydrodynamics (MHD) equations. This third-order CESE method preserves all the favorable attributes of the original second-order CESE method, such as: (i) flux conservation in space and time without using an approximated Riemann solver, (ii) genuine multi-dimensional algorithm without dimensional splitting, (iii) the use of the most compact mesh stencil, involving only the immediate neighboring cells surrounding the cell where the solution at a new time step is sought, and (iv) an explicit, unified space-time integration procedure without using a quadrature integration procedure. In order to verify the accuracy and efficiency of the scheme, several 2D MHD test problems are presented. The result of MHD smooth wave problem shows third-order convergence of the scheme. The results of the other MHD test problems show that the method can enhance the solution quality by comparing with the original second-order CESE scheme.