Mission-critical industrial infrastructure, such as data centers, increasingly depends on complex management software. Its operations, however, pose significant challenges due to the escalating system complexity, multi-vendor integration, and a shortage of expert operators. While Robotic Process Automation (RPA) offers partial automation through handcrafted scripts, it suffers from limited flexibility and high maintenance costs. Recent advances in Large Language Model (LLM)-based graphical user interface (GUI) agents have enabled more flexible automation, yet these general-purpose agents face five critical challenges when applied to industrial management, including unfamiliar element understanding, precision and efficiency, state localization, deployment constraints, and safety requirements. To address these issues, we propose InfraMind, a novel exploration-based GUI agentic framework specifically tailored for industrial management systems. InfraMind integrates five innovative modules to systematically resolve different challenges in industrial management: (1) systematic search-based exploration with virtual machine snapshots for autonomous understanding of complex GUIs; (2) memory-driven planning to ensure high-precision and efficient task execution; (3) advanced state identification for robust localization in hierarchical interfaces; (4) structured knowledge distillation for efficient deployment with lightweight models; and (5) comprehensive, multi-layered safety mechanisms to safeguard sensitive operations. Extensive experiments on both open-source and commercial DCIM platforms demonstrate that our approach consistently outperforms existing frameworks in terms of task success rate and operational efficiency, providing a rigorous and scalable solution for industrial management automation.