This paper presents a control solution for the problem of slung load transportation, with simultaneous damping of oscillations, using an underactuated autonomous quadrotor vehicle in the presence of external disturbances. The strategy put forward is divided into two parts: (i) resorting to an error definition that encompasses the pendular motion of the suspended load, a controller inspired in sliding-mode theory is developed to compute a vectored thrust actuation that actively damps out the system's oscillations; (ii) two different approaches are illustrated for the stage of attitude tracking: one stemming from the conventional backstepping technique, which controls the vehicle in angular velocity commands; and another, also based on sliding-mode theory, and built intrinsically in (Formula presented.), that produces torque commands. The overall control scheme is proved to be robust and stable, with the load oscillations shown to be damped out in the process. Simulation and experimental results are presented in a trajectory tracking scenario to validate and assess the robustness and efficiency of the proposed technique.