This paper proposes a receiver for direct-sequence spread spectrum transmissions in underwater acoustic channels, which combines a per-survivor processing (PSP) structure with sparse channel estimation. Specifically, the PSP structure establishes the trellis on the symbol level to render a small to moderate number of states, thus reducing the computational complexity. Meanwhile, the sparse channel estimation is performed on the chip level, where the orthogonal matching pursuit algorithm is used and a two-dimensional grid of path delay and Doppler scaling factor is incorporated in the dictionary construction. The effective combination of the PSP detection and sparse channel estimation achieves a good tradeoff between performance and complexity. Simulation and experiment results show that the proposed receiver outperforms the conventional RAKE receiver considerably, and most importantly, the proposed PSP receiver with an exact wideband dictionary maintains an excellent performance even for challenging underwater acoustic channels with large Doppler disparities on different paths.