We present an algorithm for estimating higher-order statistical moments of multidimensional functions expressed as polynomial chaos expansions (PCE). The algorithm starts by decomposing the PCE into a low-rank tensor network using a combination of tensor-train and Tucker decompositions. It then efficiently calculates the desired moments in the compressed tensor domain, leveraging the highly linear structure of the network. Using three benchmark engineering functions, we demonstrate that our approach offers substantial speed improvements over alternative algorithms while maintaining a minimal and adjustable approximation error. Additionally, our method can calculate moments even when the input variable distribution is altered, incurring only a small additional computational cost and without requiring retraining of the regressor.