Due to the often limited communication bandwidth of edge devices, most existing federated learning (FL) methods randomly select only a subset of devices to participate in training at each communication round. Compared with engaging all the available clients, such a random-selection mechanism could lead to significant performance degradation on non-IID (independent and identically distributed) data. In this paper, we present our key observation that the essential reason resulting in such performance degradation is the class-imbalance of the grouped data from randomly selected clients. Based on this observation, we design an efficient heterogeneity-aware client sampling mechanism, namely, Federated Class-balanced Sampling (Fed-CBS), which can effectively reduce class-imbalance of the grouped dataset from the intentionally selected clients. We first propose a measure of class-imbalance which can be derived in a privacy-preserving way. Based on this measure, we design a computation-efficient client sampling strategy such that the actively selected clients will generate a more class-balanced grouped dataset with theoretical guarantees. Experimental results show that Fed-CBS outperforms the status quo approaches in terms of test accuracy and the rate of convergence while achieving comparable or even better performance than the ideal setting where all the available clients participate in the FL training.