A new method for reducing the cost of propagating uncertainty in hypersonic flow simulations due to physical modeling parameters is proposed. This method uses a sensitivity gradient formulation, similar to a Newton method, to quickly compute new solutions to the Navier-Stokes equations given a small perturbation to the nominal input parameters. This makes the method well suited to doing probabilistic uncertainty propagation, where many solutions must be computed. There are numerical stability issues associated with using this method, and they are explored, and possible remedies suggested. The applicability of this method to uncertainty propagation is demonstrated with a simple test case.