We study the two-dimensional, time-dependent hydrodynamics of radiation-driven winds from luminous accretion disks threaded by a strong, large-scale, ordered magnetic field. The radiation force is mediated primarily by spectral lines and is calculated using a generalized multidimensional formulation of the Sobolev approximation. The effects of the magnetic field are approximated by adding into the equation of motion a force that emulates a magnetocentrifugal force. Our approach allows us to calculate disk winds when the magnetic field controls the geometry of the flow, forces the flow to corotate with the disk, or both. In particular, we calculate models in which the lines of the poloidal component of the field are straight and inclined to the disk at a fixed angle. Our numerical calculations show that flows that conserve specific angular velocity have larger mass-loss rates than their counterparts that conserve specific angular momentum. The difference in the mass-loss rates between these two types of winds can be several orders of magnitude for low disk luminosities but vanishes for high disk luminosities. Winds that conserve angular velocity have much higher velocities than angular momentum-conserving winds. Fixing the wind geometry stabilizes winds that are unsteady when the geometry is derived self-consistently. The inclination angle between the poloidal velocity and the normal to the disk midplane is important. Nonzero inclination angles allow the magnetocentrifugal force to increase the mass-loss rate for low luminosities and increase the wind velocity for all luminosities. The presence of the azimuthal force does not change the mass-loss rate when the geometry of the flow is fixed. Our calculations also show that the radiation force can launch winds from magnetized disks. The line force can be essential in producing magnetohydrodynamical (MHD) winds from disks where the thermal energy is too low to launch winds or where the field lines make an angle of less than 30° with respect to the normal to the disk midplane. In the latter case the wind will be less decollimated by the centrifugal force near its base, and its collimation far away from the disk can be larger than the collimation of its centrifugally driven MHD counterpart.