Large language models (LLMs) are increasingly prevalent in conversational systems due to their advanced understanding and generative capabilities in general contexts. However, their effectiveness in task-oriented dialogues (TOD), which requires not only response generation but also effective dialogue state tracking (DST) within specific tasks and domains, remains less satisfying. In this work, we propose a novel approach FNCTOD for solving DST with LLMs through function calling. This method improves zero-shot DST, allowing adaptation to diverse domains without extensive data collection or model tuning. Our experimental results demonstrate that our approach achieves exceptional performance with both modestly sized open-source and also proprietary LLMs: with in-context prompting it enables various 7B or 13B parameter models to surpass the previous state-of-the-art (SOTA) achieved by ChatGPT, and improves ChatGPT's performance beating the SOTA by 5.6% Avg. JGA. Individual model results for GPT-3.5 and GPT-4 are boosted by 4.8% and 14%, respectively. We also show that by fine-tuning on a small collection of diverse task-oriented dialogues, we can equip modestly sized models, specifically a 13B parameter LLaMA2-Chat model, with functioncalling capabilities and DST performance comparable to ChatGPT while maintaining their chat capabilities. We will open-source experimental code and model.