Unknown measurement delays usually degrade system performance, and even damage the system under output feedback control, which motivates us to develop an effective method to attenuate or offset the adverse effect from the measurement delays. In this paper, an augmented observer is proposed for discrete-time Lipschitz nonlinear systems subjected to unknown measurement delays, enabling a simultaneous estimation for system states and perturbed terms caused by the output delays. On the basis of the estimates, a sensor compensation technique is addressed to remove the influence from the measurement delays to the system performance. Furthermore, an integrated robust estimation and compensation technique is proposed to decouple constant piece-wise disturbances, attenuate other disturbances/noises, and offset the adverse effect caused by the measurement delays. The proposed methods are applied to a two-stage chemical reactor with delayed recycle, and an electro-mechanical servo system, which demonstrates the effectiveness of the present techniques.