Autonomous surgery has attracted increasing attention for revolutionizing robotic patient care, yet remains a distant and challenging goal. In this paper, we propose an image-based framework for high-precision autonomous suturing operation. We first build an algebraic geometric algorithm to achieve accurate needle pose estimation, then design the corresponding keypoint-based calibration network for joint-offset compensation, and further plan and control suture trajectory. Our solution ranked first among all competitors in the AccelNet Surgical Robotics Challenge. Videos and codes can be found in https://sites.google.com/view/accel-2022-cuhk.