We establish a quantitative Cauchy theory in Sobolev spaces for the Maxwell-Stefan equations with an incompressibility-like condition on the total flux. More precisely, we prove global existence and uniqueness of strong solutions, and their exponential trend to equilibrium in a perturbative regime around any macroscopic equilibrium state of the mixture, not necessarily constant. In particular, an orthogonal viewpoint that we found specific to the incompressible setting, combined with the use of a suitable anisotropic norm, allows us to get rid of the usual closure assumption of equimolar diffusion which is here recovered as an intrinsic feature of the model.