Business process automation (BPA) plays a crucial role in business process management (BPM) by improving efficiency, reducing costs, and increasing flexibility. While traditional rule-based robotic process automation (RPA) effectively handles repetitive processes, it struggles with cognitive tasks that require reasoning and decision-making. Large Language Models (LLMs) have emerged as a promising solution for cognitive automation, offering reasoning capabilities and adaptability to complex, unstructured processes. However, current LLM agent-based automation approaches focus on executing user instructions, lacking mechanisms for process reusability and adherence to structured workflows, akin to RPA systems. To address this gap, we propose an LLM agent-based automation approach that integrates process rules to enhance process adherence and reusability. Our approach incorporates a rule database that allows the LLM agent to execute known processes, easily adapt to new processes by extending its rules, and flexibly handle user requests. Additionally, we introduce a structured rule-generation mechanism to improve process-related reasoning and reduce errors. We evaluate our approach in a simulation using 118 real-world process instances from a power grid provider, comparing LLM automation with and without structured process rules. Our findings demonstrate that integrating process rules significantly improves automation success rates, reducing process failures from 16% to 1% and minimizing erroneous decision-making and hallucinations, providing a significant step towards cognitive automation while enabling the integration with diverse IT systems, BPA, and RPA approaches.