A (n + 1)-dimensional Einstein-Gauss-Bonnet (EGB) model is considered. For diagonal cosmological-type metrics, the equations of motion are reduced to a set of Lagrange equations. The effective Lagrangian contains two "minisuperspace" metrics on R^n. The first one is the well-known 2-metric of pseudo-Euclidean signature and the second one is the Finslerian 4-metric that is proportional to n-dimensional Berwald-Moor 4-metric. When a "synchronous-like" time gauge is considered the equations of motion are reduced to an autonomous system of first-order differential equations. For the case of the "pure" Gauss-Bonnet model, two exact solutions with power-law and exponential dependence of scale factors (with respect to "synchronous-like" variable) are obtained. (In the cosmological case the power-law solution was considered earlier in papers of N. Deruelle, A. Toporensky, P. Tretyakov and S. Pavluchenko.) A generalization of the effective Lagrangian to the Lowelock case is conjectured. This hypothesis implies existence of exact solutions with power-law and exponential dependence of scale factors for the "pure" Lowelock model of m-th order.