This paper proposes an extrapolation-RELAX estimator based on spectrum partitioning (SP) for the direction of arrival (DOA) estimation of frequency-modulated continuous-wave (FMCW) radar. The FMCW radar employs fast Fourier transform (FFT)-based digital beamforming (DBF) for the DOA estimation owing to its low complexity and easy implementation. However, the DBF algorithm has a disadvantage of low angle resolution. To improve the angle resolution, super-resolution algorithms such as multiple signal classification (MUSIC) and estimation of signal parameters via rotational invariance techniques (ESPRIT) are proposed. However, these algorithms require the high signal-to-noise ratio (SNR) to meet the required performance. To overcome this drawback of super-resolution algorithms, the SP-based extrapolation method has been proposed. However, this algorithm still has the problem that the resolution performance degrades owing to the insufficient number of actual antenna arrays. To solve this problem, we propose the SP-based extrapolation-RELAX algorithm for DOA estimation of FMCW radar. Through extrapolation, the proposed structure solves the problem of insufficient number of arrays, resulting in high reliability of SP results. When the extrapolation algorithm is used to generate the input signal of the RELAX algorithm, the RELAX method improves the performance of the DOA estimation. To confirm the effectiveness of the proposed estimation, we compare the Monte Carlo simulation and root-mean-square error results of the proposed and conventional algorithms. To verify the performance of the proposed algorithm in practical conditions, experiments were performed using the FMCW radar module within a chamber and in an indoor environment.