We investigate the momentum-space entanglement entropy and spectrum of several disordered one-dimensional free-fermion systems that circumvent Anderson localization, such as the random-dimer model, after a quantum quench. We numerically observe two different types of momentum-space entanglement entropy dynamics, an interesting slow logarithmic-like growth followed by saturation or rapid saturation. The type of dynamics one observes depends on the Fermi level of the intial state and the scattering matrix element structure in momentum-space. We then discuss when the momentum-space entanglement spectrum reveals the presence of delocalized states after a quench in these systems. We find if there are vanishing momentum-scattering states, the momentum-space entanglement spectrum clearly reveals the presence of delocalized states for long times.