Limited computation and communication capabilities of clients pose significant challenges in federated learning (FL) over resource-limited edge nodes. A potential solution to this problem is to deploy off-the-shelf sparse learning algorithms that train a binary sparse mask on each client with the expectation of training a consistent sparse server mask yielding sparse weight tensors. However, as we investigate in this paper, such naive deployments result in a significant drop in accuracy compared to FL with dense models, especially for clients with limited resource budgets. In particular, our investigations reveal a serious lack of consensus among the trained sparsity masks on clients, which prevents convergence for the server mask and potentially leads to a substantial drop in model performance. Based on such key observations, we propose federated lottery aware sparsity hunting (FLASH), a unified sparse learning framework to make the server win a lottery in terms of yielding a sparse sub-model, able to maintain classification performance under highly resource-limited client settings. Moreover, to support FL on different devices requiring different parameter density, we leverage our findings to present hetero-FLASH, where clients can have different target sparsity budgets based on their device resource limits. Experimental evaluations with multiple models on various datasets (both IID and non-IID) show superiority of our models in closing the gap with unpruned baseline while yielding up to ∼ 10.1% improved accuracy with ∼ 10.26x fewer communication costs, compared to existing alternatives, at similar hyperparameter settings.