Access to prescription drugs is a critical issue for adults 65 and older, almost 90% of whom take at least one medication to manage their health. About one in four Medicare beneficiaries has difficulty affording prescription drugs, resulting in delayed and unfilled prescriptions, and the cost burden is higher among non-Hispanic Black and Latino/a adults. We used nationally representative data to evaluate how differences in socioeconomic resources, access to care, health care need, and the use of cost-saving strategies contributed to long-standing racial and ethnic inequalities in prescription cost burden among older Medicare beneficiaries. We used logistic regressions and the Karlson-Holm-Breen decomposition method to quantify the degrees to which each factor accounted for the heightened cost burden among non-Hispanic Black and Latino/a beneficiaries compared to non-Hispanic White beneficiaries. Non-Hispanic White beneficiaries had lower odds of cost burden and higher odds of engaging in cost-saving strategies compared to marginalized groups. Even after controlling for socioeconomic, insurance, and health characteristics, non-Hispanic Black beneficiaries were more likely to experience prescription drug cost burden, and Hispanic or Latino/a beneficiaries were less likely to use cost-saving strategies compared to non-Hispanic White beneficiaries. Socioeconomic inequalities accounted for a large share of the greater cost burden experienced by older non-Hispanic Black and Hispanic/Latino adults. Notably, Hispanic/Latino beneficiaries' lower use of cost-saving strategies was a significant driver of their greater cost burden. Our study highlights how Medicare, a near-universal insurance system for older Americans, still reproduces systemic inequities and threatens the health and financial well-being of many.