The growing complexity of contemporary microwave circuits made numerical optimization imperative as a performance-boosting tool. Yet, it is intricate because of the high costs incurred by electromagnetic (EM) analysis required to evaluate the system’s quality reliably. These expenses are particularly significant in global optimization, which is necessary in many situations. This study introduces an innovative strategy for high-efficacy globalized optimization of passive components. Our methodology leverages reduction of the problem dimensionality implemented using a rapid global sensitivity analysis and a custom-developed machine learning (ML) algorithm employing fast surrogate models established in the reduced domain. The designs rendered by the ML process are further refined in the local sense in the full-dimensionality parameter space using a gradient-based routine. Additional improvement in efficiency is obtained by employing multi-fidelity EM simulations with the low-fidelity models used for global search and high-fidelity ones only utilized in fine-tuning. The presented approach has been comprehensively validated utilizing two coupling circuits and juxtaposed against a pool of benchmark algorithms. The obtained results underscore the remarkable efficacy of our procedure. The typical running cost does not exceed a hundred high-fidelity EM analyses, corresponding to sizable savings over the benchmark. At the same time, the proposed method renders designs of competitive quality.