The transient response of a nonlinear multidegreeoffreedom system with viscoelasticity subjected to Gaussian white noise excitation is investigated. Firstly, the effect of the viscoelasticity on the system is approximated by the modified damping and stiffness. The original system is replaced by a system without viscoelasticity. Then, the stochastic averaging method based on generalized harmonic functions is adopted to derive the averaged FokkerPlanckKolmogorov equation of amplitude transient joint probability density for each oscillator. This equation is solved by expressing the probability density as multiple series in terms of a set of properly statedependent orthogonal basis functions with timedependent coefficients. According to Galerkin method, the timedependent coefficients can be solved from a set of firstorder linear differential equations. Finally, the semianalytical formulae of the transient probability density as well as the transient probability of the state response and the statistical moments for the amplitude response is obtained. To illustrate the proposed procedure, coupled twodegreeof freedom Duffingvan der Pol oscillators with viscoelasticity subjected to Gaussian white noise excitation is investigated as an example. The effect of viscoelasticity on the system response is initially discussed. Moreover, comparison with the simulation results of the original system indicates that the proposed procedure is accuracy and efficacy.