The nonintrusive Chebyshev interval method is used to analyze the transient response of a rotor system. The main idea is to analyze the uncertainty in speed-varying rotor systems and reduce the error in the calculation. The hybrid analysis procedure, which combines the black-box modeling and overestimation controlling, is proposed via the Chebyshev approximation and searching the extreme values of the approximation function for the uncertain response. It can both increase accuracy and improve the computational efficiency of the proposed method. Effects of the uncertain parameters on response bounds and estimation errors of the system are investigated, e.g., supporting stiffness and damping. The accuracy and efficiency of the proposed method are verified via comparing with the scanning method. Overestimation effects caused by the interval arithmetic can be controlled, particularly the sensitive domains near the critical speed. It can provide guidance for general uncertain transient mechanical systems, particularly those that have severe fluctuations near resonances.