Recent advances in large language models (LLMs) have led to the development of artificial intelligence (AI)-powered tutoring chatbots, showing promise in providing broad access to high-quality personalized education. Existing works have primarily studied how to make LLMs follow tutoring principles but not how to model student behavior in dialogues. However, analyzing student dialogue turns can serve as a formative assessment, since open-ended student discourse may indicate their knowledge levels and reveal specific misconceptions. In this work, we present a first attempt at performing knowledge tracing (KT) in tutor-student dialogues. We propose LLM prompting methods to identify the knowledge components/skills involved in each dialogue turn and diagnose whether the student responds correctly to the tutor, and verify the LLM's effectiveness via an expert human evaluation. We then apply a range of KT methods on the resulting labeled data to track student knowledge levels over an entire dialogue. We conduct experiments on two tutoring dialogue datasets, and show that a novel yet simple LLM-based method, LLMKT, significantly outperforms existing KT methods in predicting student response correctness in dialogues. We perform extensive qualitative analyses to highlight the challenges in dialogue KT and outline multiple avenues for future work.