We aim to tackle the problem of point-based interactive segmentation, in which two key challenges are to infer user's intention correctly and to propagate the user-provided annotations to unlabeled regions efficiently. To address those challenges, we propose a novel intention-aware feature propagation strategy that performs explicit user intention estimation and learns an efficient click-augmented feature representation for high-resolution foreground segmentation. Specifically, we develop a coarse-to-fine sparse propagation network for each interactive segmentation step, which consists of a coarse-level network for more effective tracking of user's interest, and a fine-level network for zooming to the target object and performing fine-level segmentation. Moreover, we design a new sparse graph network module for both levels to enable efficient long-range propagation of click information. Extensive experiments show that our method surpasses the previous state-of-the-art methods on all popular benchmarks, demonstrating its efficacy.