In this study, a deeply buried circular tunnel supported with point-anchored rock bolts was investigated. Assuming that the effects of each rock bolt on the surrounding rock are simplified to two identical concentrated forces acting in opposite directions, a viscoelastic analytical solution for the axial forces of the rock bolts is presented herein based on the complex variable method and correspondence principle. First, in the solution process, the surrounding rock is considered as an elastic material. A linear equation system for the axial forces was established according to the deformation coordination condition between the surrounding rock and the rock bolts. Second, the surrounding rock was considered as a viscoelastic material. Based on the correspondence principle, Laplace transformation was performed for the equation system of the axial forces to obtain its expression in the Laplace domain, and the values of the axial forces in the Laplace domain could be determined by solving the obtained expression. Finally, the viscoelastic semi-analytical solution for the axial forces was established using the numerical method of the Laplace inverse transformation. The results obtained were also compared with the numerical methods of ANSYS. Accordingly, the effects of the lateral stress coefficient, rock mass Young’s modulus, rock bolt length, viscosity coefficient, and preload on the axial forces of the rock bolts were discussed.