CsV_{3}Sb_{5} is a newly discovered Z_{2} topological kagome metal showing the coexistence of a charge-density-wave (CDW)-like order at T^{*}=94 K and superconductivity (SC) at T_{c}=2.5 K at ambient pressure. Here, we study the interplay between CDW and SC in CsV_{3}Sb_{5} via measurements of resistivity, dc and ac magnetic susceptibility under various pressures up to 6.6 GPa. We find that the CDW transition decreases with pressure and experience a subtle modification at P_{c1}≈0.6-0.9 GPa before it vanishes completely at P_{c2}≈2 GPa. Correspondingly, T_{c}(P) displays an unusual M-shaped double dome with two maxima around P_{c1} and P_{c2}, respectively, leading to a tripled enhancement of T_{c} to about 8 K at 2 GPa. The obtained temperature-pressure phase diagram resembles those of unconventional superconductors, illustrating an intimated competition between CDW-like order and SC. The competition is found to be particularly strong for the intermediate pressure range P_{c1}≤P≤P_{c2} as evidenced by the broad superconducting transition and reduced superconducting volume fraction. The modification of CDW order around P_{c1} has been discussed based on the band structure calculations. This work not only demonstrates the potential to raise T_{c} of the V-based kagome superconductors, but also offers more insights into the rich physics related to the electron correlations in this novel family of topological kagome metals.