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This article deals with the stochastic Gross–Pitaevskii equation (SGPE) perturbed with multiplicative time noise. The numerical solutions of the governing model are carried out with the proposed stochastic non-standard finite difference (SNSFD) scheme. The stability of the scheme is proved by using the Von-Neumann criteria and the consistency is shown in the mean square sense. To seek exact solutions, we applied the Sardar subequation (SSE) and modified exponential rational functional (MERF) techniques. The exact solutions are constructed in the form of exponential, hyperbolic, and trigonometric forms. Finally, the comparison of the exact solutions with numerical solutions is drawn in the 3D and line plots for the different values of parameters.