The direction-of-arrival (DoA) estimation algorithms have a fundamental role in target bearing estimation by sensor array systems. Recently, compressive sensing (CS)-based sparse reconstruction techniques have been investigated for DoA estimation due to their superior performance relative to the conventional DoA estimation methods, for a limited number of measurement snapshots. In many underwater deployment scenarios, the acoustic sensor arrays must perform DoA estimation in the presence of several practical problems such as unknown source number, faulty sensors, low values of the received signal-to-noise ratio (SNR), and access to a limited number of measurement snapshots. In the literature, CS-based DoA estimation has been investigated for the individual occurrence of some of these errors but the estimation under joint occurrence of these errors has not been studied. This work investigates the CS-based robust DoA estimation to account for the joint impact of faulty sensors and low SNR conditions experienced by a uniform linear array of underwater acoustic sensors. Most importantly, the proposed CS-based DoA estimation technique does not require a priori knowledge of the source order, which is replaced in the modified stopping criterion of the reconstruction algorithm by taking into account the faulty sensors and the received SNR. Using Monte Carlo techniques, the DoA estimation performance of the proposed method is comprehensively evaluated in relation to other techniques.