In acoustic-based methods for water pipeline leak localization, there are few studies on complex multi-leak localization since these sources mix with and affect each other during propagation. In this paper, a novel multi-leak localization method based on the convolutional blind source separation (CBSS) algorithm is proposed, in which the time delays are estimated by the quadrature filter of the demixing matrix or the cross-correlation function between the separated signals and the measured signals, and then the leakage locations are found by linear localization formula. In theory, this paper analyzes the basic principle of the algorithm for locating multiple leaks. In simulation results, the algorithm’s feasibility is verified. In addition, it is obtained that the location distribution of the leakage sources will affect the location of the peak of the delay, and that strong noise will reduce the applicability of the algorithm. In prototype experiments, the algorithm is verified to locate the leaks effectively. This paper is the first to apply CBSS to multi-leakage localization, providing a feasible solution to this complex problem.