In this article, we study the stochastic complex coupled Kuralay model, which possesses some applications in various fields, including physics, biology, and engineering, to obtain new solitary wave solutions. The explicit analytical solutions are obtained by using the Sardar subequation method, which helps to illuminate the dynamics of oscillators under random (noisy) effects. The integration of the Wiener process along a given method is a precise approximation of the stochastic behavior of the system. The proposed strategy enables the derivation of several exact solitary wave solutions under stochastic conditions, including bright, dark, and singular wave profiles. More significantly, the obtained solutions are also represented by 3D surface and contour plots that clearly show how solitary waves change and evolve when noise is introduced. Other stochastic models in physics and engineering can use the proposed approach to understand the workings of complex systems.