This paper proposes a control strategy integrating the non-linear extended state observer (NLESO) and the non-singular terminal sliding mode control (NTSMC) for the trajectory tracking of wheeled mobile robots subject to bounded disturbances. A new transformation method of chained model in terms of Lie derivative is presented to simplify the controller design. A specific NLESO combining linear term and non-linear term is designed to estimate the disturbances with a faster convergence performance. A scheme for determining the gain range of NLESO is explicitly given to facilitate the tuning of experimental parameters. Meanwhile, the NTSMC achieves finite time convergence of the tracking error system and the chattering phenomenon in NTSMC is dramatically alleviated with the compensation from NLESO. The experimental results validate the strong robustness and good performance of the proposed control strategy.