The non-contact detection of human vital signs (i.e., respiration rate (RR) and heartbeat rate (HR)) using a continuous-wave (CW) Doppler radar sensor has great potential for intensive care monitoring, home healthcare, etc. However, large-scale and fast random body movement (RBM) has been a bottleneck for vital sign detection using a single CW Doppler radar. To break this dilemma, this study proposed a scheme combining adaptive noise cancellation (ANC) with polynomial fitting, which could retrieve the weak components of both respiration and heartbeat signals that were submerged under serious RBM interference. In addition, the new-type discrete cosine transform (N-DCT) was introduced to improve the detection accuracy. This scheme was first verified using a numerical simulation. Then, experiments utilizing a 10-GHz Doppler radar sensor that was built from general-purpose radio frequency (RF) and communication instruments were also carried out. No extra RF/microwave components and modules were needed, and neither was a printed circuit board nor an integrated-chip design required. The experimental results showed that both the RR and HR could still be extracted during large-scale and fast body movements using only a single Doppler radar sensor because the RBM noises could be greatly eliminated by utilizing the proposed ANC algorithm.