We address 3D path- and motion-planning in cluttered environments for a quadrotor UAV using a 3D depth sensor with limited range and field-of-view. The 3D elevation map of obstacles in the environment is assumed a priori unknown, but a 2D projection of this map onto a horizontal plane is assumed a priori fully known. We specifically address path-planning to discover and traverse “3D shortcuts,” namely, to achieve reductions in path costs using 3D depth sensor information, as compared to planning using the 2D projection map alone. We propose a 3D path-planning algorithm based on incremental repair of a constant-altitude seed path computed from the 2D map. Two strategies are employed to expedite the discovery of such repairs. We demonstrate the effectiveness of the proposed algorithm via a high-fidelity numerical simulation, which includes the proposed path-planning algorithm, dynamically feasible trajectory generation, trajectory tracking, and simulation of the depth sensor.