Feature extraction on point clouds is an essential task when analyzing and processing point clouds of 3D scenes. However, there still remains a challenge to adequately exploit local fine-grained features on point cloud data due to its irregular and unordered structure in a 3D space. To alleviate this problem, a Dilated Graph Attention-based Network (DGANet) with a certain feature for learning ability is proposed. Specifically, we first build a local dilated graph-like region for each input point to establish the long-range spatial correlation towards its corresponding neighbors, which allows the proposed network to access a wider range of geometric information of local points with their long-range dependencies. Moreover, by integrating the dilated graph attention module (DGAM) implemented by a novel offset–attention mechanism, the proposed network promises to highlight the differing importance on each edge of the constructed local graph to uniquely learn the discrepancy feature of geometric attributes between the connected point pairs. Finally, all the learned edge attention features are further aggregated, allowing the most significant geometric feature representation of local regions by the graph–attention pooling to fully extract local detailed features for each point. The validation experiments using two challenging benchmark datasets demonstrate the effectiveness and powerful generation ability of our proposed DGANet in both 3D object classification and segmentation tasks.