This study presents an analytic sparse-view cone-beam computed tomography (CBCT) reconstruction method using a novel sinogram restoration technique based on higher-order Fourier harmonic peaks to obtain CBCT images of reasonable quality with a reduced radiation dose. The proposed method involves four main steps: (1) acquiring sparse-view (P144) sinogram from a CBCT system, (2) upsampling (P720) the original sparse-view sinogram with equally spaced zero padding, (3) separating higher-order Fourier harmonic peaks of the upsampled sinogram using a band-pass filter, and (4) restoring the sinogram of the higher-order harmonic peaks by taking its magnitude, followed by computationally cost-efficient filtered-backprojection (FBP)-based CBCT reconstruction and denoising processes. The notation P144 indicates that the number of projections is 144. To verify the efficacy of the proposed method, an experiment was performed on a chest phantom using a prototype CBCT system comprising an X-ray tube (90 kV p and 40 mA), a large-area flat-panel detector (388 μm pixel size), and a rotational gantry. The quality of the restored sparse-view CBCT images was quantitatively evaluated in terms of the image intensity profile, universal quality index (UQI), and peak signal-to-noise ratio (PSNR). The results indicate that the proposed sinogram restoration method effectively recovered zero-padded areas in the original upsampled sinogram with image intensities nearly identical to those of the reference dense-view (P720) sinogram, which demonstrates the efficacy of the proposed sinogram restoration method. The quality of the sparse-view CBCT image restored with higher-order harmonic peaks was close to that of the reference dense-view CBCT image, maintaining the image quality. The UQI and PSNR values of the restored sparse-view CBCT image were 0.93 and 34.84, respectively, which are approximately 1.3 and 1.2 times greater than those of the original sparse-view CBCT image, respectively, improving the image quality. Consequently, the proposed method has the potential to resolve the issues caused by sparse-view sampling in FBP-based CBCT.