Underwater acoustic signals (UAS) in marine environments are often contaminated by noise from biological activity, ocean waves, and equipment. Under low signal-to-noise ratio (SNR) conditions, target ship-radiated components and background noise are strongly mixed, making reliable underwater detection, recognition, and communication challenging. To address this problem, this paper proposes Multi-Scale TCN, DPRNN, and Attention TasNet (MSDA-TasNet), a lightweight end-to-end source separation network for underwater ship-radiated noise. The proposed method combines time-domain convolutional encoding, a dual-path recurrent neural network (DPRNN), and a multi-scale temporal convolutional network (multi-scale TCN) to model both short-term details and long-term dependencies of UAS. To further enhance target-aware feature representation, this paper designs a novel channel–frequency attention mechanism that adaptively emphasizes target-related spectral cues. Experiments on two underwater acoustic datasets show that MSDA-TasNet achieves better separation performance than several representative baselines in terms of Signal-to-Distortion Ratio (SDR), Scale-Invariant Signal-to-Distortion Ratio (SI-SDR), SDR improvement (SDRi), SI-SDR improvement (SI-SDRi), and Segmental Signal-to-Distortion Ratio (SegSDR). The results indicate that the proposed network can improve target signal recovery under complex underwater acoustic noise while maintaining a relatively low parameter count and computational cost.