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Most existing black-box optimization methods assume that all variables in the system being optimized have equal cost and can change freely at each iteration. However, in many real world systems, inputs are passed through a sequence of different operations or modules, making variables in earlier stages of processing more costly to update. Such structure imposes a cost on switching variables in early parts of a data processing pipeline. In this work, we propose a new algorithm for switch cost-aware optimization called Lazy Modular Bayesian Optimization (LaMBO). This method efficiently identifies the global optimum while minimizing cost through a passive change of variables in early modules. The method is theoretical grounded and achieves vanishing regret when augmented with switching cost. We apply LaMBO to multiple synthetic functions and a three-stage image segmentation pipeline used in a neuroscience application, where we obtain promising improvements over prevailing cost-aware Bayesian optimization algorithms. Our results demonstrate that LaMBO is an effective strategy for black-box optimization that is capable of minimizing switching costs in modular systems.