Modeling the kinematics and dynamics of robotics systems with suspended loads using dual quaternions has not been explored so far. This paper introduces a new innovative control strategy using dual quaternions for UAVs with cable-suspended loads, focusing on the sling load lifting and tracking problems. By utilizing the mathematical efficiency and compactness of dual quaternions, a unified representation of the UAV and its suspended load's dynamics and kinematics is achieved, facilitating the realization of load lifting and trajectory tracking. The simulation results have tested the proposed strategy's accuracy, efficiency, and robustness. This study makes a substantial contribution to present this novel control strategy that harnesses the benefits of dual quaternions for cargo UAVs. Our work also holds promise for inspiring future innovations in under-actuated systems control using dual quaternions.