Large language models (LLMs) excel in generating fluent utterances but can lack reliable grounding in verified information. At the same time, knowledge-graph-based fact-checkers deliver precise and interpretable evidence, yet suffer from limited coverage or latency. By integrating LLMs with knowledge graphs and real-time search agents, we introduce a hybrid fact-checking approach that leverages the individual strengths of each component. Our system comprises three autonomous steps: 1) a Knowledge Graph (KG) Retrieval for rapid one-hop lookups in DBpedia, 2) an LM-based classification guided by a task-specific labeling prompt, producing outputs with internal rule-based logic, and 3) a Web Search Agent invoked only when KG coverage is insufficient. Our pipeline achieves an F1 score of 0.93 on the FEVER benchmark on the Supported/Refuted split without task-specific fine-tuning. To address Not enough information cases, we conduct a targeted reannotation study showing that our approach frequently uncovers valid evidence for claims originally labeled as Not Enough Information (NEI), as confirmed by both expert annotators and LLM reviewers. With this paper, we present a modular, opensource fact-checking pipeline with fallback strategies and generalization across datasets.