Infrared small target detection (ISTD) is essential for early warning and remote sensing applications. However, this task remains challenging due to severe background clutter and low signal-to-noise ratios. Although deep unfolding networks combine model-driven interpretability with data-driven learning, existing methods based on low-rank and sparse decomposition often rely on global assumptions and fail to capture high-frequency local gradient information, resulting in blurred target boundaries and increased false alarm rates. To address this issue, we propose a novel model-based framework: the Deep Gradient Unfolding Network (DGraNet). We incorporate an Adaptive Gradient Structure Prior (AGSP) into the Maximum A Posteriori (MAP) framework to capture local target saliency, and unfold the optimization process into a deep neural network using the Alternating Direction Method of Multipliers (ADMM). Additionally, we introduce a learnable gradient estimation module to approximate complex proximal operators, enabling adaptive edge refinement and effective clutter suppression. Experiments on the NUDT-SIRST, SIRST-Aug, and IRSTD-1k datasets demonstrate that DGraNet achieves state-of-the-art performance, with improved structural preservation and enhanced robustness under low signal-to-noise ratio conditions.