Extracting depth information from photon-limited, defocused images is challenging because depth from defocus (DfD) relies on accurate estimation of defocus blur, which is fundamentally sensitive to image noise. We present a novel approach to robustly measure object depths from photon-limited images along the defocused boundaries. It is based on a new image patch representation, Blurry-Edges, that explicitly stores and visualizes a rich set of low-level patch information, including boundaries, color, and smoothness. We develop a deep neural network architecture that predicts the Blurry-Edges representation from a pair of differently defocused images, from which depth can be calculated using a closed-form DfD relation we derive. The experimental results on synthetic and real data show that our method achieves the highest depth estimation accuracy on photon-limited images compared to a broad range of state-of-the-art DfD methods.