Ontology matching (OM) enables semantic interoperability between different ontologies and resolves their conceptual heterogeneity by aligning related entities. OM systems currently have two prevailing design paradigms: conventional knowledge-based expert systems and newer machine learning-based predictive systems. While large language models (LLMs) and LLM-based agents have become revolutionary in data engineering and have been applied creatively in various domains, their potential for OM remains underexplored. This study introduces a novel agent-powered LLM-based design paradigm for OM systems. With thoughtful consideration of several specific challenges to leverage LLMs for OM, we propose a generic framework, namely Agent-OM, consisting of two Siamese agents for retrieval and matching, with a set of simple prompt-based OM tools. Our framework is implemented in a proof-of-concept system. Evaluations of three Ontology Alignment Evaluation Initiative (OAEI) tracks over state-of-the-art OM systems show that our system can achieve very close results to the best long-standing performance on simple OM tasks and significantly improve the performance on complex and few-shot OM tasks.