Sampling rate synchronization is an inevitable issue in distributed acoustic sensor networks. In this paper, an analytical sampling rate offset (SRO) estimation approach is first proposed, and then, it is extended to a distributed method that suitable for acoustic sensor networks with arbitrary communication graphs. Specifically, a linear-phase drift model in the short-time Fourier transform domain is used to approximate the SRO between each pair of microphone nodes. Next, after unwrapping the temporally averaged phase information, SROs are recovered analytically via a new weighted-sum criterion. Based on this, a distributed cost function is established at each node to obtain the SROs of all nodes simultaneously in a distributed manner. Finally, a state-of-the-art distributed algorithm named asynchronous network Newton optimization is adopted to carry out the distributed SRO estimation. The proposed method can effectively estimate the SROs among acoustic sensor nodes in noisy and reverberant environments. Compared with the existing approaches, it does not require an external central processor, and only local communications among nodes are needed. Experimental results confirm the validity of the proposed method.