The kagome metal CsV₃Sb₅ hosts an intriguing interplay between charge-density-wave (CDW) order and superconductivity that is highly sensitive to lattice distortions. However, determining the specific roles of the in-plane (A_(1g,1)) and out-of-plane (A_(1g,2)) symmetric strain channels has been hindered by their intrinsic mixing in conventional piezo-based experiments. Here, we combine in-plane uniaxial strain with direct c-axis compression to independently access and disentangle these symmetry-resolved responses in CsV₃Sb₅. We reveal that c-axis compression drives a massive, linear enhancement of the superconducting transition temperature (T_c) alongside a suppression of T_(rm CDW). The tuning efficiency of this out-of-plane deformation acts with an opposite sign and far exceeds that of in-plane strain, demonstrating that c-axis lattice control dictates the phase competition. Furthermore, by isolating the pure elastoresistivity coefficients, we find that the out-of-plane cross-coupling coefficient (m₁₃) is comparable in magnitude but opposite in sign to the in-plane response (m₁₁+m₁₂). Unlike the sharply peaked in-plane response, m₁₃ exhibits a distinct, order-parameter-like onset across the CDW transition. Our results establish that out-of-plane lattice control plays a dominant role in tuning the intertwined states in CsV₃Sb₅ and provide a general pathway for resolving strain-coupled electronic responses in layered quantum materials.