We study dust models in higher-dimensional general relativity and the Einstein–Gauss–Bonnet theory in a maximally symmetric spacetime. We show that the dynamics of the model is governed by an Abel differential equation of the second kind. This equation is solved in general for all arbitrary dimensions and spatial curvature to yield solutions for the cosmic scale factor. We show that the existence of the first integral determining the dynamics of dust can be demonstrated without additional assumptions. Analysis of the first integral shows that explicit and implicit solutions are possible depending on the presence of the cosmological constant. Comparisons between the higher-order theory and general relativity are discussed in detail.