Observability in cloud infrastructure is critical for service providers, driving the widespread adoption of anomaly detection systems for monitoring metrics. However, existing systems often struggle to simultaneously achieve explainability, reproducibility, and autonomy, which are three indispensable properties for production use. We introduce Argos, an agentic system for detecting time-series anomalies in cloud infrastructure by leveraging large language models (LLMs). Argos proposes to use explainable and reproducible anomaly rules as intermediate representation and employs LLMs to autonomously generate such rules. The system will efficiently train error-free and accuracy-guaranteed anomaly rules through multiple collaborative agents and deploy the trained rules for low-cost online anomaly detection. Through evaluation results, we demonstrate that Argos outperforms state-of-the-art methods, increasing $F_1$ scores by up to $9.5\%$ and $28.3\%$ on public anomaly detection datasets and an internal dataset collected from Microsoft, respectively.