After the structural model is discretized by the finite element method, it may introduce false high-frequency components that can adversely affect the structural dynamic response of the structure. Therefore, it is necessary to introduce numerical damping into the integration algorithm to effectively suppress these false high-frequency componments. Based on a semi-explicit integration formulation, this paper proposes an unconditionally stable semi-explicit integration algorithm with controllable numerical damping by matching the characteristic equation coefficients of the amplification matrix of the implicit ρ∞-Bathe algorithm. The new semi-explicit (NSE)-ρ∞ algorithm controls the numerical damping of the integration algorithm by two free coefficients ρ∞ and γ, and does not require weighted equation of motion. The numerical characteristics of the new algorithm, such as stability, accuracy, period elongation and amplitude decay, are analyzed. It is found that the new algorithm is unconditionally stable for both linear elastic and nonlinear stiffness softening systems. Through representative numerical examples, the new algorithm is compared with two other unconditionally stable explicit integration algorithms with controllable numerical damping. The analytical results demonstrate that the new algorithm is more effective in suppressing the spurious high-frequency components.