The article investigates an optimal control problem described by retarded differential inclusions (DFIs) of higher order with endpoint constraints. In terms of the Euler-Lagrange type adjoint inclusions and the Hamiltonian, a sufficient optimality condition is derived for higher-order DFIs that have different forms in different time intervals depending on the delay parameter, which in the problem without delay, these two adjoint inclusions coincide. It is shown that the adjoint inclusion for the DFIs of the first order, defined in terms of locally conjugate mappings, coincides with the classical Euler-Lagrange inclusion. Thus, the obtained results are universal in the sense that sufficient optimality conditions can be formulated for a DFI of any order. At the end of the paper, problems with a high-order polyhedral DFIs and higher-order linear optimal control problems are considered, the optimality conditions of which are transformed into the Pontryagin maximum principle. Also, for highorder polyhedral optimization with delay, from the point of view of abstract economics, non-negative adjoint variables can be interpreted as the price of a resource.