Dynamical systems in engineering and physics are often subject to irregular excitations that are best modeled as random. Monte Carlo simulations are routinely performed on such random models to obtain statistics on their long-term response. Such simulations, however, are prohibitively expensive and time consuming for high-dimensional nonlinear systems. Here we propose to decrease this numerical burden significantly by reducing the full system to very low-dimensional, attracting, random invariant manifolds in its phase space and performing the Monte Carlo simulations on that reduced dynamical system. The random spectral submanifolds (SSMs) we construct for this purpose generalize the concept of SSMs from deterministic systems under uniformly bounded random forcing. We illustrate the accuracy and speed of random SSM reduction by computing the SSM-reduced power spectral density of the randomly forced mechanical systems that range from simple oscillator chains to finite-element models of beams and plates.