We study the two-dimensional, time-dependent magnetohydrodynamics (MHD) of radiation-driven winds from luminous accretion disks initially threaded by a purely axial magnetic field. The radiation force is mediated primarily by spectral lines and is calculated using a generalized multidimensional formulation of the Sobolev approximation. We use ideal MHD to compute numerically the evolution of Keplerian disks, varying the magnetic field strengths and the luminosity of the disk, the central accreting object, or both. We find that the magnetic fields very quickly start deviating from purely axial because of the magnetorotational instability. This leads to fast growth of the toroidal magnetic field as field lines wind up because of the disk rotation. As a result the toroidal field dominates over the poloidal field above the disk and the gradient of the former drives a slow and dense disk outflow, which conserves specific angular momentum. Depending on the strength of the magnetic field relative to the system luminosity, the disk wind can be radiation or MHD driven. The pure radiation-driven wind consists of a dense, slow out flow that is bounded on the polar side by a high-velocity stream. The mass-loss rate is mostly due to the fast stream. As the magnetic field strength increases, first the slow part of the flow is affected; namely, it becomes denser and slightly faster and begins to dominate the mass-loss rate. In very strong magnetic field or pure MHD cases, the wind consists of only a dense, slow out flow without the presence of the distinctive fast stream so typical of pure radiation-driven winds. Our simulations indicate that winds launched by the magnetic fields are likely to remain dominated by the fields downstream because of their relatively high densities. The radiation force due to lines may not be able to change a dense MHD wind because the line force strongly decreases with increasing density.