Self-assembled DNA nanostructures show great promise as functional devices, highly configurable materials, and in nanorobotics. Magnetic control can provide a powerful actuation mechanism in a broad range of contexts, since it affords a high-level of external control, it is biocompatible, and orthogonal to chemical or electrical stimuli. Here we demonstrate magnetic molecular nanoactuators by leveraging the unique site-specificity of DNA origami to assemble highly anisotropic magnetic nanocubes on high-aspect ratio DNA origami bundles. We traced and controlled 100s of our DNA origami nanorotors at the single-rotor level and demonstrated their programmable magnetic clamping and controlled rotation under uniform and rotating magnetic fields. By varying the population and inter-particle spacing of the nanocubes, magnetic torque values in the order of 10-100 pN nm are achieved at field strengths < 10 mT. Monte Carlo simulations reveal that assembly of nanocubes on DNA origami rotors leads to collective magnetic properties, with numerically estimated torque values in good agreement with the experiments. Our magnetic nanorotors offer a foundation for biocompatible nanorobotics, as well as high-throughput magnetic force and torque tweezers.