The fourth order three component partial differential equations of vorticity transport in terms of three-dimensional stream function vector components are formulated. The generalized Galerkin finite element solutions of the governing equations for incompressible flow is then carried out, and the results are shown to be very accurate for the lid-driven three-dimensional cavity problems examined herein. This numerical accuracy is attributed to the correct definition of three-dimensional stream function vector components, appropriate finite element interpolation functions, associated physically simple boundary conditions, and the rigorous Newton-Raphson scheme of solution process, as well as the governing equations being free from pressure oscillations. To the best of our knowledge, the present study marks the first attempt to solve the three-dimensional fourth order partial differential equations of voilicity transport for the three-dimensional stream function vector components.