A method which uses only the velocity components as primitive variables is described for solution of the incompressible unsteady Navier-Stokes equations. The method involves the multiplication of the primitive variable-based Navier-Stokes equations with the unit normal vector of finite volume elements and the integration of the resulting equations along the boundaries of four-node quadrilateral finite volume elements. Therefore, the pressure term is eliminated from the governing equations and any difficulty associated with pressure or vorticity boundary conditions is avoided. The equations are discretized on four-node quadrilateral finite volume elements by using the second-order-accurate central finite differences with the mid-point integral rule in space and the first-order-accurate backward finite differences in time. The resulting system of algebraic equations is solved in coupled form using a direct solver. As a test case, an impulsively accelerated lid-driven cavity flow in a square enclosure is solved in order to verify the accuracy of the present method.