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A novel robust controller is proposed in this study to realize the precise motion control of a cell puncture mechanism (CPM) driven by piezoelectric ceramics (PEAs). The entire dynamic model of CPM is constructed based on the Bouc–Wen model, and the nonlinear part of the dynamic model is optimized locally to facilitate the construction of a robust controller. A model-based, nonlinear robust controller is constructed using time-delay estimation (TDE) and fractional-order nonsingular terminal sliding mode (FONTSM). The proposed controller does not require prior knowledge of unknown disturbances due to its real-time online estimation and compensation of unknown terms by using the TDE technology. The controller also hasfinite-time convergence and high-precision trajectory tracking capabilities due to FONTSM manifold and fast terminal sliding mode-type reaching law. The stability of the closed-loop system is proved by Lyapunov stability theory. Computer simulation and hardware-in-loop simulation experiments of CPM verify that the proposed controller outperforms traditional terminal sliding mode controllers, such as the integer-order or model-free controller. The proposed controller can also continuously output without chattering and has high control accuracy. Zebrafish embryo is used as a verification target to complete the cell puncture experiment. From the engineering application perspective, the proposed control strategy can be effectively applied in a PEA-driven CPM.