The aim of this work is to propose and analyse a new high-order discontinuous Galerkin finite element method for the time integration of a Cauchy problem for second-order ordinary differential equations. These equations typically arise after space semidiscretization of second-order hyperbolic-type differential problems, e.g., wave, elastodynamics and acoustics equations. After introducing the new method, we analyse its well-posedness and prove a priori error estimates in a suitable (mesh-dependent) norm. Numerical results are also presented to verify our theoretical estimates.