In this paper, an exponentially stable trajectory tracking controller is proposed for multi-UAV payload transport. The multi-UAV payload system has a 2-DOF magnetic spherical joint between the UAVs and the vertical rigid links of the payload frame, so the UAVs can roll or pitch freely. These vertical links are rigidly attached to the payload and cannot move. An input-output feedback linearized model is derived for the complete payload-UAV system along with thrust vectoring control for trajectory tracking of the payload. The theoretical analysis on tracking control laws shows that control law is exponentially stable, thus guaranteeing safe transportation along the desired trajectory. To validate the performance of the proposed control law, the results for a numerical simulation as well as a high-fidelity Gazebo real-time simulation are presented. Next, the robustness of the proposed controller is analyzed against two practical situations: External disturbance on the payload and payload mass uncertainty. The results clearly indicate that the proposed controller is robust and computationally efficient while achieving exponentially stable trajectory tracking.