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Control systems are inevitably affected by external disturbances, and a major objective of the control design is to attenuate or eliminate their adverse effects on the system performance. This paper presents a disturbance rejection approach with two main improvements over existing results: 1) it relaxes the requirement of calculating or measuring the state derivatives, which are not available for measurement, and their calculation is corrupted by noise, and 2) it achieves finite-time disturbance rejection and control. To this end, the disturbance is first modeled by an unknown dynamics, and an adaptive disturbance observer is proposed to estimate it. A filtered regressor form is leveraged to model the nonlinear system and the unknown disturbance. It is shown that using this filtered regressor form, the disturbance is estimated using only measured state of the regressor. That is, contrary to the existing results on disturbance rejection, the presented approach does not require the state derivative measurements. To improve the convergence speed of the disturbance estimation, an adaptive law, equipped with experience replay, is presented. The disturbance observer is then augmented with an adaptive integral terminal sliding mode control to assure the finite-time convergence of tracking error to zero. A verifiable rank condition on the history of the past experience used by the experience-replay technique provides a sufficient condition for convergence. Compared to the existing results, neither the knowledge of the disturbance dynamics nor the state derivatives are required, and finite-time stability is guaranteed. A simulation example illustrates the effectiveness of the proposed approach.