Computational argumentation offers formal frameworks for transparent, verifiable reasoning but has traditionally been limited by its reliance on domain-specific information and extensive feature engineering. In contrast, LLMs excel at processing unstructured text, yet their opaque nature makes their reasoning difficult to evaluate and trust. We argue that the convergence of these fields will lay the foundation for a new paradigm: Argumentative Human-AI Decision-Making. We analyze how the synergy of argumentation framework mining, argumentation framework synthesis, and argumentative reasoning enables agents that do not just justify decisions, but engage in dialectical processes where decisions are contestable and revisable -- reasoning with humans rather than for them. This convergence of computational argumentation and LLMs is essential for human-aware, trustworthy AI in high-stakes domains.