Automated scoring plays a crucial role in education by reducing the reliance on human raters, offering scalable and immediate evaluation of student work. While large language models (LLMs) have shown strong potential in this task, their use as end-to-end raters faces challenges such as low accuracy, prompt sensitivity, limited interpretability, and rubric misalignment. These issues hinder the implementation of LLM-based automated scoring in assessment practice. To address the limitations, we propose AutoSCORE, a multi-agent LLM framework enhancing automated scoring via rubric-aligned Structured COmponent REcognition. With two agents, AutoSCORE first extracts rubric-relevant components from student responses and encodes them into a structured representation (i.e., Scoring Rubric Component Extraction Agent), which is then used to assign final scores (i.e., Scoring Agent). This design ensures that model reasoning follows a human-like grading process, enhancing interpretability and robustness. We evaluate AutoSCORE on four benchmark datasets from the ASAP benchmark, using both proprietary and open-source LLMs (GPT-4o, LLaMA-3.1-8B, and LLaMA-3.1-70B). Across diverse tasks and rubrics, AutoSCORE consistently improves scoring accuracy, human-machine agreement (QWK, correlations), and error metrics (MAE, RMSE) compared to single-agent baselines, with particularly strong benefits on complex, multi-dimensional rubrics, and especially large relative gains on smaller LLMs. These results demonstrate that structured component recognition combined with multi-agent design offers a scalable, reliable, and interpretable solution for automated scoring.