Grasping of diverse objects in unstructured environments remains a significant challenge. Open-loop grasping methods, effective in controlled settings, struggle in cluttered environments. Grasp prediction errors and object pose changes during grasping are the main causes of failure. In contrast, closed-loop methods address these challenges in simplified settings (e.g., single object on a table) on a limited set of objects, with no path to generalization. We propose Grasp-MPC, a closed-loop 6-DoF vision-based grasping policy designed for robust and reactive grasping of novel objects in cluttered environments. Grasp-MPC incorporates a value function, trained on visual observations from a large-scale synthetic dataset of 2 million grasp trajectories that include successful and failed attempts. We deploy this learned value function in an MPC framework in combination with other cost terms that encourage collision avoidance and smooth execution. We evaluate Grasp-MPC on FetchBench and real-world settings across diverse environments. Grasp-MPC improves grasp success rates by up to 32.6% in simulation and 33.3% in real-world noisy conditions, outperforming open-loop, diffusion policy, transformer policy, and IQL approaches. Videos and more at http://grasp-mpc.github.io.