The Kelvin constitutive model is modified by the spring-pot element based on the Caputo fractional derivative to describe the mechanical behavior of one-dimensional consolidation of saturated soil. After introducing the continuous drainage boundary condition, the analytical solutions of the effective stress and the settlement under time-dependent loading are derived by performing Laplace transformation. The Laplace inverse transformation is used to obtain the theoretical solutions in time domain, and the influences of relevant parameters on the settlement under trapezoidal cyclic loading and construction loading are studied. The results show that the settlement of viscoelastic soil under cyclic loading increases in an oscillating manner, and the amplitude of the oscillation increases with the boundary permeability. A higher value of the fractional order α slows the development of settlement in the early stage of consolidation. However, in the later stage of consolidation, the effect of α on settlement is reversed. The oscillation amplitude of the settlement under cyclic loading decreases with increase of α. Furthermore, detailed analysis indicates that the development of one-dimensional consolidation settlement is also related to mechanical properties of soil and loading parameters. The larger the elastic modulus E is, the smaller the final settlement, and the greater the delay time of viscoelastic is, the slower the settlement occurs.