Gene drives—artificial genetic constructs that bias their own inheritance, enabling them to spread through a population—are promising tools for controlling disease vectors and invasive species. While gene drives can potentially eradicate target populations, their unintended spillover into non-target populations poses substantial ecological concerns. Here, we develop a general methodological framework to examine how integrating gene drives with established pest-control methods, such as pesticide application and sterile-male release, can mitigate spillover risk. In this framework, we develop an optimal-control model that identifies treatment strategies with minimum total costs of eradication, accounting for direct treatment expenditures and potential damage from spillover. We show that pesticides are most cost-effective at the intermediate stage of eradication when gene drives have already slowed population growth but do not yet substantially suppress the population. For sterile-male release, however, we observe two distinct types of optimal strategies, with the optimal choice depending on parameter values: intensive early-stage releases or mild later-stage releases. Using established pest controls also relaxes requirements for gene drive design, favoring milder suppressive effects as the treatment intensifies. Although spillover risk is hard to quantify in real-world contexts, the proposed approach identifies the maximum relative reduction in the spillover-risk attainable with a given level of treatment effort.