Post-tensioned rocking systems have proved to be highly effective in controlling structural damage during strong ground motions. However, recent events have highlighted the importance of looking at both the structural and non-structural components within a holistic framework. In this context, the high rotations and accelerations associated with the rocking motion can cause significant non-structural damage and affect the performance and functionality of the entire system. In this paper, we analytically examine the fundamental dynamics of post-tensioned rocking structures and investigate the benefits of using supplemental rotational inertia to reduce their seismic demands and improve their overall performance. The newly proposed strategy employs inerters, a mechanical device that develops a resisting force proportional to the relative acceleration between its terminals. Analyses conducted for a wide range of acceleration pulses and real pulse-like ground motions show that post-tensioned structures equipped with inerters consistently experience lower demands and have reduced probabilities of exceeding limit states typically associated with damage. Importantly, the new vibration control strategy advanced in this paper opens the door for an expedient modification of the fundamental dynamic response of rocking systems without altering their geometry.