In this paper a problem of stationary flow of generalized Newtonian fluid in a thin channel is considered. An efficient algorithm of solution is proposed that includes a flexible procedure for a continuous approximation of the apparent viscosity by means of elementary functions combined with analytical integration of the governing equations. The algorithm can be easily adapted to circular or elliptic conduits. The accuracy and efficiency of computations are analyzed using an example of the Carreau fluid. The proposed computational scheme proves to be highly efficient and versatile providing excellent accuracy of solution at a very low computational cost.