In this paper, a fifth-order weighted essentially nonoscillatory scheme is presented for simulating dam-break flows in a finite difference framework. The new scheme is a convex combination of two quadratic polynomials with a fourth-degree polynomial in a classical WENO fashion. The distinguishing feature of the present method is that the same five-point information is used but smaller absolute truncation errors and the same accuracy order in the smooth region are obtained. The new nonlinear weights are presented by Taylor expansion of the smoothness indicators of the small stencils to sustain the optimal fifth-order accuracy. The linear advection equation, nonlinear scalar Burgers equation, and one- and two-dimensional Euler equations are used to validate the high-order accuracy and excellent resolution of the presented method. Finally, one- and two-dimensional Saint-Venant equations are tested by using the new fifth-order scheme to simulate a dam-break flow.