Abstract Understanding the competition between superconductivity and other ordered states (such as antiferromagnetic or charge-density-wave (CDW) state) is a central issue in condensed matter physics. The recently discovered layered kagome metal A V 3 Sb 5 ( A  = K, Rb, and Cs) provides us a new playground to study the interplay of superconductivity and CDW state by involving nontrivial topology of band structures. Here, we conduct high-pressure electrical transport and magnetic susceptibility measurements to study CsV 3 Sb 5 with the highest T c of 2.7 K in A V 3 Sb 5 family. While the CDW transition is monotonically suppressed by pressure, superconductivity is enhanced with increasing pressure up to P1 ≈ 0.7 GPa, then an unexpected suppression on superconductivity happens until pressure around 1.1 GPa, after that, T c is enhanced with increasing pressure again. The CDW is completely suppressed at a critical pressure P2 ≈ 2 GPa together with a maximum T c of about 8 K. In contrast to a common dome-like behavior, the pressure-dependent T c shows an unexpected double-peak behavior. The unusual suppression of T c at P1 is concomitant with the rapidly damping of quantum oscillations, sudden enhancement of the residual resistivity and rapid decrease of magnetoresistance. Our discoveries indicate an unusual competition between superconductivity and CDW state in pressurized kagome lattice.