Learning performant robot manipulation policies can be challenging due to high-dimensional continuous actions and complex physics-based dynamics. This can be alleviated through intelligent choice of action space. Operational Space Control (OSC) has been used as an effective task-space controller for manipulation. Nonetheless, its strength depends on the underlying modeling fidelity, and is prone to failure when there are modeling errors. In this work, we propose OSC for Adaptation and Robustness (OSCAR), a data-driven variant of OSC that compensates for modeling errors by inferring relevant dynamics parameters from online trajectories. OSCAR decomposes dynamics learning into task-agnostic and task-specific phases, decoupling the dynamics dependencies of the robot and the extrinsics due to its environment. This structure enables robust zero-shot performance under out-of-distribution and rapid adaptation to significant domain shifts through additional finetuning. We evaluate our method on a variety of simulated manipulation problems, and find substantial improvements over an array of controller baselines.