We consider a wide class of randomly perturbed systems subjected to stationary Gaussian processes and show that chaotic orbits exist almost surely under some degenerate condition, no matter how small the random forcing terms are. This result is very contrast to the deterministic forcing case, in which chaotic orbits exist only if the influence of the forcing terms overcomes that of the other terms in the perturbations. To obtain the result, we extend Melnikov's method and prove that the corresponding Melnikov functions, which we call the Melnikov processes, have infinitely many zeros, so that infinitely many transverse homoclinic orbits exist. In addition, a stable and unstable manifold theorem is given and the Smale-Birkhoff homoclinic theorem is extended in appropriate forms for randomly perturbed systems. We illustrate our theory for the randomly perturbed Duffing oscillator subjected to the Ornstein-Uhlenbeck process.