Interacting with object instances is crucial in specific embodied tasks such as rescue and industrial services, not just navigating to the target. Previous research on Object Goal Navigation (ObjNav) has only addressed navigating to the object’s vicinity. However, the interface between approaching the object and finely localizing and grasping it has yet to be adequately investigated. This paper presents the Finding and Grasping (FaG) framework, which is based on the mobile manipulator, to address the last-mile ObjNav problem. The framework incorporates an object localization module and a sub-target extraction module into a traditional autonomous exploration and navigation pipeline. The object localization module uses a grasp estimation method based on point cloud completion to determine the global goal. In another module, a sub-target extraction algorithm is used to break down global goals into navigation and grasping sub-goals. The proposed framework effectively solves the ObjNav last-mile problem and is superior to the baseline in all settings, as demonstrated by sufficient experiments. Furthermore, real robot grasping experiments were conducted to verify the method’s feasibility. A demo video is available at: https://youtu.be/AGlIO-DBM1g