Circularly polarized (CP) γ-ray sources are versatile for broad applications in nuclear physics, high-energy physics, and astrophysics. The laser-plasma based particle accelerators provide accessibility for much higher flux γ-ray sources than conventional approaches, in which, however, the circular polarization properties of the emitted γ-photons are usually neglected. In this Letter, we show that brilliant CP γ-ray beams can be generated via the combination of laser plasma wakefield acceleration and plasma mirror techniques. In a weakly nonlinear Compton scattering scheme with moderate laser intensities, the helicity of the driving laser can be transferred to the emitted γ-photons, and their average polarization degree can reach ∼61% (20%) with a peak brilliance of ≳1021 photons/(s · mm2 · mrad2 · 0.1% BW) around 1 MeV (100 MeV). Moreover, our proposed method is easily feasible and robust with respect to the laser and plasma parameters.