We study the Ginzburg dg algebra Γ_T associated with the quiver with potential arising from a triangulation T of a decorated marked surface S_△, in the sense of [22]. We show that there is a canonical way to identify all finite-dimensional derived categories D_{fd}(Γ_T), denoted by D_{fd}(S_△). As an application, we show that the spherical twist group ST(S_△) associated with D_{fd}(S_△) acts faithfully on its space of stability conditions.