We present a novel framework designed to extend model reconciliation approaches, commonly used in human-aware planning, for enhanced human-AI interaction. By adopting a structured argumentation-based dialogue paradigm, our framework enables dialectical reconciliation to address knowledge discrepancies between an explainer (AI agent) and an explainee (human user), where the goal is for the explainee to understand the explainer's decision. We formally describe the operational semantics of our proposed framework, providing theoretical guarantees. We then evaluate the framework's efficacy ``in the wild'' via computational and human-subject experiments. Our findings suggest that our framework offers a promising direction for fostering effective human-AI interactions in domains where explainability is important.