Universal grasping with multi-fingered dexterous hands is a fundamental challenge in robotic manipulation. While recent approaches successfully learn closed-loop grasping policies using reinforcement learning (RL), the inherent difficulty of high-dimensional, long-horizon exploration necessitates complex reward and curriculum design, often resulting in suboptimal solutions across diverse objects. We propose DemoGrasp, a simple yet effective method for learning universal dexterous grasping. We start from a single successful demonstration trajectory of grasping a specific object and adapt to novel objects and poses by editing the robot actions in this trajectory: changing the wrist pose determines where to grasp, and changing the hand joint angles determines how to grasp. We formulate this trajectory editing as a single-step Markov Decision Process (MDP) and use RL to optimize a universal policy across hundreds of objects in parallel in simulation, with a simple reward consisting of a binary success term and a robot-table collision penalty. In simulation, DemoGrasp achieves a 95% success rate on DexGraspNet objects using the Shadow Hand, outperforming previous state-of-the-art methods. It also shows strong transferability, achieving an average success rate of 84.6% across diverse dexterous hand embodiments on six unseen object datasets, while being trained on only 175 objects. Through vision-based imitation learning, our policy successfully grasps 110 unseen real-world objects, including small, thin items. It generalizes to spatial, background, and lighting changes, supports both RGB and depth inputs, and extends to language-guided grasping in cluttered scenes.