Knowledge graphs represent real-world facts as structured triplets and underpin a wide range of applications, including question answering, recommendation, and retrieval-augmented generation. Automatically extracting such triplets from unstructured text is essential for scalable knowledge base construction. Traditional extraction methods require task-specific training data and struggle to generalize across domains. Large language models (LLMs) offer an alternative through in-context learning, enabling flexible extraction without fine-tuning. However, LLMs frequently hallucinate—generating plausible triplets unsupported by the source text. The root cause is the lack of provenance: existing methods produce triplets without explicit links to their textual origins, making faithfulness unverifiable. This paper presents Anchor-Extraction-Verification-Supplement (AEVS), a framework that grounds every triplet element to the source text. AEVS operates in three stages: (1) anchor discovery identifies entities, relation phrases, and attribute values with precise positions, forming a constrained extraction vocabulary; (2) grounded extraction generates triplets linked to discovered anchors; and (3) restoration-based verification validates triplets through hierarchical matching, with a coverage-aware supplement ensuring comprehensive extraction. Experiments on WebNLG, REBEL, and Wiki-NRE demonstrate consistent improvements over both trained models and LLM-based baselines. Ablation studies confirm that anchor-based constraints are the primary mechanism for hallucination reduction. Dedicated analyses of anchor discovery quality, computational cost (2.83–4.28 LLM calls per sample), and hallucination rates (0.23–20.23% across model–dataset configurations) provide insights into the framework’s practical applicability and limitations.