Sawtooth events in the Earth's magnetosphere are global, large-amplitude oscillations of energetic plasma particle fluxes at geosynchronous orbit and represent periodic magnetospheric substorms with a typical period of similar to 3 hours. Sawtooth events generally occur during magnetic storms, when the magnetosphere is continuously driven by southward interplanetary magnetic field and high-speed solar wind stream. However, it has not been well understood how the magnetotail parameters (the total pressure, the lobe magnetic field, and the tail lobe total magnetic flux) at the onset of sawtooth events are related to the solar wind driver. In this study, we conduct a statistical analysis of the magnetotail parameters measured by the Geotail satellite during sawtooth events over 1998-2006. At the onset of sawtooth events (storm-time substorms), the magnetotail total pressure and the lobe magnetic field increase with the solar wind pressure and merging electric field, and the total magnetic flux in the tail lobe increases with the merging electric field. Empirical formulas of the relationship of the magnetotail parameters at the sawtooth onset and the solar wind are derived for the first time. We have made a superposed epoch analysis. The magnetotail total pressure and lobe magnetic field take 52 min for gradual buildup and then 26 min for rapid increase before the sawtooth onset, and they decrease for 77 min after the onset. We have also compared our results with previous studies on quiet time tail behavior and isolated substorms. The magnetotail total pressure at the sawtooth onset is about three times that of the quiet time magnetotail, and the lobe magnetic field at the sawtooth onset is 8-10 nT higher than the quiet time value. The results imply that the sawtooth onset occurs when the magnetotail reaches a critical state and that the critical state depends on the solar wind parameters. Our findings provide new insights into the storm-time magnetospheric dynamics and important guidance for model simulations.