Secure aggregation is a popular protocol in privacy-preserving federated learning, which allows model aggregation without revealing the individual models in the clear. On the other hand, conventional secure aggregation protocols incur a significant communication overhead, which can become a major bottleneck in real-world bandwidth-limited applications. Towards addressing this challenge, in this work we propose a lightweight gradient sparsification framework for secure aggregation, in which the server learns the aggregate of the sparsified local model updates from a large number of users, but without learning the individual parameters. Our theoretical analysis demonstrates that the proposed framework can significantly reduce the communication overhead of secure aggregation while ensuring comparable computational complexity. We further identify a trade-off between privacy and communication efficiency due to sparsification. Our experiments demonstrate that our framework reduces the communication overhead by up to 7.8x, while also speeding up the wall clock training time by 1.13x, when compared to conventional secure aggregation benchmarks.