The paper develops a control procedure for a noisy near-Hamiltonian system. The control task is to prevent the system from escape from a reference region bounded by a lobe of the separatrix. Motion near the separatrix is presented as a sequence of encirclements over the separatrix lobe between two consecutive vertices, and the problem of avoiding escape through the separatrix is reduced to maximization of energy losses during one encirclement. It is shown that the energy difference during an encirclement can be approximated by the stochastic Melnikov integral. This allows extension of the stochastic Melnikov method to optimal control problems. An approximate solution is constructed as a time-invariant feedback, which is proved to be a nearly-optimal control for the original nonstationary problem.