We propose a new dimension-wise splitting with selective relaxation (DSSR) method for saddle point systems arising from the discretization of the incompressible Navier-Stokes equations. Using Fourier analysis, we determine the optimal choice of the relaxation parameter that leads to the best perform ance of the iterative method for the Stokes and the steady Oseen equations. We also explore numerically the influence of boundary conditions on the optimal choice of the parameter, the use of inner and outer iterations, and the perform ance for a lid driven cavity flow.