Based on significant improvements in engineering materials, three advanced engineering measures have been proposed—super anchor cables, high-strength concrete anti-fault caverns, and grouting modification using high-strength concrete-to resist fault dislocation in the surrounding rock near tunnels crossing active strike-slip faults. Moreover, single- or multiple-joint advanced engineering measures form the local rock mass-anti-fault (LRAF) method. A numerical method was used to investigate the influence of LRAF methods on the stress and displacement fields of the surrounding rock, and the anti-fault effect was evaluated. Finally, the mechanism of action of the anchor cable was verified using a three-dimensional numerical model. The numerical results indicated that the anchor cable and grouting modification reduced the displacement gradient of the local surrounding rock near the tunnels crossing fault. Furthermore, anchor cable and grouting modifications changed the stress field of the rock mass in the modified area. The tensile stress field of the rock mass in the modified anchor cable area was converted into a compressive stress field. The stress field in the modified grouting area changed from shear stress in the fault slip direction to tensile stress in the axial tunnel direction. The anti-fault cavern resisted the dislocation displacement and reduced the maximum dislocation magnitude, displacement gradient, and shear stress. Among the three advanced engineering measures, the anchor cable was the core of the three advanced engineering measures. An anchor cable, combined with other LRAF measures, can form an artificial safety island at the cross-fault position of the rock mass to protect the tunnel. The research results provide a new supporting idea for the surrounding rock of tunnels crossing active strike-slip faults.