In this work, we present SuFIA, the first framework for natural language-guided augmented dexterity for robotic surgical assistants. SuFIA incorporates the strong reasoning capabilities of large language models (LLMs) with perception modules to implement high-level planning and low-level control of a robot for surgical sub-task execution. This enables a learning-free approach to surgical augmented dexterity without any in-context examples or motion primitives. SuFIA uses a human-in-the-loop paradigm by restoring control to the surgeon in the case of insufficient information, mitigating unexpected errors for mission-critical tasks. We evaluate SuFIA on four surgical sub-tasks in a simulation environment and two sub-tasks on a physical surgical robotic platform in the lab, demonstrating its ability to perform common surgical sub-tasks through supervised autonomous operation under challenging physical and workspace conditions. Project website: orbit-surgical.github.io/sufia