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In this paper, a new algorithm for high-resolution multiple wideband and nonstationary source localization using a sensor array is proposed. The received signals of the sensor array are first converted into the time-frequency domain via short-time Fourier transform (STFT) and we find that a set of short-time power spectrum matrices at different time instants have the joint diagonalization structure in each frequency bin. Based on such joint diagonalization structure, a novel cost function is designed and a new spatial spectrum for direction-of-arrival (DOA) estimation at hand is derived. Compared to the maximum-likelihood (ML) method with high computational complexity, the proposed algorithm obtains the DOA estimates via one-dimensional (1-D) search instead of multidimensional search. Therefore its computational complexity is much lower than the ML method. Unlike the subspace-based high-resolution DOA estimation techniques, it is not necessary to determine the number of sources in advance for the proposed algorithm. Moreover, the proposed method is robust to the effects of reverberation caused by multipath reflections. Hence it is suitable for multiple acoustic source localization in a reverberant room. The results of numerical simulations and experiments in a real room with a moderate reverberation are provided to demonstrate the good performance of the proposed approach.