In this paper, an effective oscillation-free discontinuous Galerkin (DG) scheme for a nonlinear stochastic convection-dominated problem is formulated and analyzed. The proposed oscillation-free scheme is capable to capture the steep fronts of solution automatically and distinguish the influence of the convection domination and noise perturbation. Under proper regularity assumptions, the optimal convergence rates in space and time are rigorously proved with the techniques of variational solution and conditional expectation. In the numerical simulation, the classical SIPG scheme and the proposed oscillation-free DG scheme are both performed and compared. The numerical convergence rates tests are first carried out to verify the theoretical results. The benchmark tests having the steep behaviors are further provided to illustrate the effectiveness and robustness of our proposed oscillation-free DG scheme.