We propose the traditional goal of cancer therapists to develop a single drug or drug combination that can, by itself, eliminate all cancer cells within a host has neglected potential treatments that may achieve curative outcomes by strategically combining agents that are individually effective but non-curative. We derive basic principles for such an approach from the eco-evolutionary dynamics of background extinctions in which a “first strike” reduces the size and heterogeneity of the initial population and is followed immediately by demographic and ecological “second strikes” that push the population below an extinction threshold. This proposed strategy appears identical to the empirically-derived curative therapy in childhood Acute Lymphocytic Leukemia.