High-accuracy Time of Arrival (TOA) estimation is a crucial requirement for indoor acoustic localization. However, conventional approaches to TOA estimation based on Generalized Cross Correlation (GCC) suffer from insufficient resolution in severe multipath environments, which hinders their ability to achieve robust ranging. In this paper, we propose a novel and robust acoustic TOA estimation method based on Frequency Modulated Continuous Wave (FMCW) that selects the first-path signal. Our method transforms multipath signals into the frequency domain using several novel signal processing techniques of FMCW, which enhances the resolution of multipath identification and makes the estimation more stable. Simulation and experimental results show that, compared to other methods, our method can achieve centimeter-level TOA ranging estimation over a distance range of 16 m. Furthermore, our proposed method can enhance the accuracy of TOA estimation-based ranging by approximately 4.2 cm in non-line-of-sight (NLOS) scenarios.