The dynamic response and isolation performance of a Quasi-Zero-Stiffness (QZS) vibration isolator using inclined springs as negative stiffness correctors under random excitation are presented in this paper. The probabilistic linearization method is employed to determine the dynamic response of the QZS vibration isolator and compared with the exact solution based on the Fokker-Planck-Kolmogorov (FPK) equation and the equivalent linearization method. Two performance indexes (Mean Square Relative Displacement (MSRD) and Mean Square Acceleration (MSA)) are considered to evaluate the isolation performance of the QZS vibration isolator under random excitation and compared with the equivalent linear vibration isolator. The results show that the MSRD of the QZS vibration isolator is always lower than the equivalent linear vibration isolator, while the MSA of the QZS vibration isolator can be larger or lower than the equivalent linear vibration isolator based on the values of damping ratio and spectral density of the random excitation.