In this paper we report the effect of uniaxial strains applied along the crystalline a axis on the newly discovered kagome superconductor CsV3Sb5. At ambient conditions,CsV3Sb5 shows a charge-density wave (CDW) transition at T-CDW = 94.5 K and superconducts below T-c = 3.34 K. In our paper, when the uniaxial strains is varied from -0.90% to 0.90%, T-c monotonically increases by similar to 33% from 3.0 to 4.0 K, giving rise to the empirical relation T-c (epsilon) = 3.4 + 0.56 epsilon + 0.12 epsilon(2) . On the other hand, fors changing from -0.76% to 1.26%, T-CDW decreases monotonically by similar to 10% from 97.5 to 87.5 K with T-CDW (epsilon) = 94.5 - 4.72 epsilon - 0.60 epsilon(2). The opposite response of T-c and T-CDW to the uniaxial strain suggests strong competition between these two orders. Comparison with hydrostatic pressure measurements indicate that it is the change in the c axis that is responsible for these behaviors of the CDW and superconducting transitions, and that the explicit breaking of the sixfold rotational symmetry by strain has a negligible effect. Combined with our first-principles calculations and phenomenological analysis, we conclude that the enhancement in T-c with decreasing c is caused primarily by the suppression of T-CDW, rather than strain-induced modifications in the bare superconducting parameters. We propose that the sensitivity of (c) with respect to the changes in the c axis arises from the impact of the latter on the trilinear coupling between the M-1(+) and the L-2(-) phonon modes associated with the CDW. Overall, our paper reveals that the c-axis lattice parameter, which can be controlled by both pressure and uniaxial strain, is a powerful tuning knob for the phase diagram of CsV3Sb5.