In this letter we apply dynamical system methods to study all evolutional paths admissible for all initial conditions of the FRW cosmological model with a non-minimally coupled to gravity scalar field and a barotropic fluid. We choose ``energy variables'' as phase variables. We reduce dynamics to a 3-dimensional dynamical system for an arbitrary potential of the scalar field in the phase space variables (κ/6^1/2H, κV^1/2/3^1/2H, κφ/6^1/2). After postulating the potential parameter Γ as a function of λ (defined as −V′/V) we reduce whole dynamics to a 3-dimensional dynamical system and study evolutional paths leading to current accelerating expansion. If we restrict the form of the potential then we will obtain a 2-dimensional dynamical system. We use the dynamical system approach to find a new generic quintessence scenario of approaching to the de Sitter attractor which appears only for the case of non-vanishing coupling constant.