This paper studies a new preconditioning technique for sparse systems arising from discretized partial differential equations (PDEs) in computational fluid dynamics (CFD), which exploit the multilevel sequentially semiseparable (MSSS) structure of the system matrix. MSSS matrix computations give a data-sparse way to approximate the LU factorization of a sparse matrix from discretized PDEs in linear computational complexity with respect to the problem size. In contrast to the standard block preconditioners, we exploit the global MSSS structure of the 2 by 2 block system from the discretized Stokes equation and linearized Navier-Stokes equation. This avoids the approximation of the Schur complement, which is a big advantage over standard block preconditioners. Numerical experiments on standard CFD benchmark problems in IFISS were carried out to evaluate the performance of the MSSS preconditioners. It was illustrated that the MSSS preconditioner yields mesh size independence convergence. Moreover, the convergence is almost insensitive to the viscosity parameter. Comparison with the algebraic multigrid (AMG) method and the geometric multigrid (GMG) method, the MSSS preconditioning technique is more robust than both the AMG method and the GMG method, and considerably faster than the AMG method.