We study the Boltzmann equation for the 3D kinetic Couette flow describing the motion of a rarefied gas confined by two parallel infinite plates moving tangentially relative to each other with opposite velocities. Specifically, we focus on the Boltzmann equation in the diffusive limit in a 3D periodic channel domain with the diffusive reflection boundary condition imposed on the plates and an external force present. Using a perturbation approach, we establish the existence of the stationary solution to the steady problem and further obtain the exponential asymptotic stability for the time-dependent problem near the obtained stationary 3D kinetic Couette flow. This result implies that the first-order approximation of the Boltzmann solutions for small Knudsen numbers is determined by the perturbed incompressible Navier-Stokes-Fourier system with the external force around the Couette flow in fluid dynamics. A key feature of our approach is to construct solutions in the Lagrangian frame with the bulk velocity for the fluid Couette flow connecting the same velocities of two moving plates. As a consequence, it appears that the shearing force makes solutions have only a polynomial tail for large velocities.