In this paper, we study the global well-posedness of the stochastic S-KdV system in $H^1(\mathbb{R})\times H^1(\mathbb{R})$, which are driven by additive noises. It is difficult to show the global well-posedness of a related perturbation system even for smooth datum and stochastic forces. To overcome it, we introduce a new sequence of approximation equations, which is the key of this paper. We establish priori estimates, global well-posedness and convergences of these approximation equations, which help us to get a pathwise priori estimate of the initial system.