We present and analyze a set of three-dimensional, global, general relativistic radiation magnetohydrodynamic simulations of thin, radiation-pressure-dominated accretion disks surrounding a nonrotating, stellar-mass black hole. The simulations are initialized using the Shakura–Sunyaev model with a mass accretion rate of M ̇ = 3 L Edd / c 2 (corresponding to L = 0.17L Edd). Our previous work demonstrated that such disks are thermally unstable when accretion is driven by an α-viscosity. In the present work, we test the hypothesis that strong magnetic fields can both drive accretion through magnetorotational instability and restore stability to such disks. We test four initial magnetic field configurations: (1) a zero-net-flux case with a single, radially extended set of magnetic field loops (dipole), (2) a zero-net-flux case with two radially extended sets of magnetic field loops of opposite polarity stacked vertically (quadrupole), (3) a zero-net-flux case with multiple radially concentric rings of alternating polarity (multiloop), and (4) a net-flux, vertical magnetic field configuration (vertical). In all cases, the fields are initially weak, with a gas-to-magnetic pressure ratio ≳100. Based on the results of these simulations, we find that the dipole and multiloop configurations remain thermally unstable like their α-viscosity counterpart, in our case collapsing vertically on the local thermal timescale and never fully recovering. The vertical case, on the other hand, stabilizes and remains so for the duration of our tests (many thermal timescales). The quadrupole case is intermediate, showing signs of both stability and instability. The key stabilizing factor is the ability of specific field configurations to build up and sustain strong, P mag ≳ 0.5P tot, toroidal fields near the midplane of the disk. We discuss the reasons why certain configurations are able to do this effectively and others are not. We then compare our stable simulations to the standard Shakura–Sunyaev disk.