This paper introduces a tensor singular value decomposition (TSVD) approach for estimating non-Gaussian Structural Vector Autoregressive (SVAR) models. The proposed methodology applies to both complete and partial identification of structural shocks. The estimation procedure relies on third- and/or fourth-order cumulants. We establish the asymptotic distribution of the estimator and conduct a simulation study to evaluate its finite-sample performance. The results demonstrate that the estimator is highly competitive in small samples compared to alternative methods under complete identification. In cases of partial identification, the estimator also exhibits very good performance in small samples. To illustrate the practical relevance of the procedure under partial identification, two empirical applications are presented.