Neural surface reconstruction has been dominated by implicit representations with marching cubes for explicit surface extraction. However, those methods typically require high-quality normals for accurate reconstruction. We propose OffsetOPT, a method that reconstructs explicit surfaces directly from 3D point clouds and eliminates the need for point normals. The approach comprises two stages: first, we train a neural network to predict surface triangles based on local point geometry, given isometrically distributed input points. Next, we apply the frozen network to reconstruct surfaces from unseen point clouds by optimizing a per-point offset to maximize the accuracy of triangle predictions. Compared to state-of-the-art methods, OffsetOPT not only excels at reconstructing overall surfaces but also significantly preserves sharp surface features. We demonstrate its accuracy on popular benchmarks, including small-scale shapes and large-scale open surfaces.