To enhance the vital sign radar's detection accuracy of heartbeat rate (HR), which is heavily affected by the harmonics of respiration frequency, this paper proposes a waveform-driven matched filtering method based on polynomial fitting extraction. The merit of this approach lies in that, the quasi-ideal matching impulse approximating the actual heartbeat signal can be readily retrieved by subtracting the polynomial fitted waveform from a received signal, which provides high adaptability for individual subjects. Essentially, this extraction greatly removes the harmonic interference originated from respiration, thus considerably improving the HR detection accuracy. Simulations are performed to acquire the proper fitting order and the effective impulse duration. Guided by these two parameters, a 10-GHz non-contact continuous-wave (CW) Doppler radar system with typical microwave laboratory instruments is constructed. Experimental results confirmed the effectiveness of the proposed method, showing that the average errors of HR can be reduced from 46.21% to 0.96% under different subjects and distances, and from 38.05% to 0.90% in continuous measurement for one subject.