In this paper, the distributed average consensus problem in sensor networks with limited data rate communication is studied. Unlike standard average consensus, only quantized signals with finite support are adopted for the communications among agents. To tackle this problem, a novel distributed algorithm is proposed, where each agent iteratively updates a local estimate based on quantized signals received by its neighbors. The proposed algorithm differs from the existing schemes dealing with limited data rate in the following key features: 1) each agent is not required to have information on spectral properties of the graph associated with the communication topology; 2) the initial measurements are not required to be bounded within a known interval; and 3) exact consensus to the average can be achieved asymptotically for weight-balanced directed topology. Thus, it is more favorable for practical implementations, especially for large networks. The proposed algorithm is proved to achieve average consensus asymptotically, almost surely and in mean square sense. The analysis of convergence rate and generalizations to random weight-balanced directed topologies and time-varying quantization are also provided. Finally, numerical results validate our theoretical findings, and demonstrate the superior performance of the proposed algorithm compared to existing topology-agnostic consensus schemes with limited data rate.