Load-bearing biological materials employ specialized elastic bridging regions to connect material parts with substantially different properties. While such bridging regions emerge in diverse systems of biological systems, their functional-mechanical origins are yet disclosed. Here, we hypothesize that these elastic bridging regions evolved primarily to minimize the near-interface stress effects in the biological material and, supported by experiments and simulations, we develop a simple theoretical model for such stress-minimizing bridging modulus. Our theoretical model describes well extensive experimental data of diverse biomechanical systems, suggesting that despite their compositionally distinct bridging regions, they share a similar mechanical adaptation strategy for stress minimization. The theoretical model developed in this study may directly serve as a design guideline for bio-inspired materials, biomedical applications, and advanced interfacial architectures with high resilience to mechanical failure. STATEMENT OF SIGNIFICANCE: Biological materials exhibit unconventional structural-mechanical strategies allowing them to attain extreme load-bearing capabilities. Here, we identify the strategy of biological materials to connect parts of distinct elastic properties in an optimal manner of stress minimization. Our findings are compatible with broad types of biological materials, including biopolymers, biominerals, and their bio-composite combinations, and may promote novel engineering designs of advanced biomedical and synthetic materials.