Full-scale dome structures intrinsically have numerous sources of irreducible aleatoric uncertainties. A large-scale numerical simulation of the dome structure is required to quantify the effects of these sources on the dynamic performance of the structure using the finite element method (FEM). To reduce the heavy computational burden, a surrogate model of a dome structure was constructed to solve this problem. The dynamic global sensitivity of elastic and elastoplastic structures was analyzed in the uncertainty quantification framework using fully quantitative variance- and distribution-based methods through the surrogate model. The model considered the predominant sources of uncertainty that have a significant influence on the performance of the dome structure. The effects of the variables on the structural performance indicators were quantified using the sensitivity index values of the different performance states. Finally, the effects of the sample size and correlation function on the accuracy of the surrogate model as well as the effects of the surrogate accuracy and failure probability on the sensitivity index values are discussed. The results show that surrogate modeling has high computational efficiency and acceptable accuracy in the uncertainty quantification of large-scale structures subjected to earthquakes in comparison to the conventional FEM.