Conventionally, a single-channel synthetic aperture radar (SC-SAR) system can hardly detect weak moving targets simply. In this paper, a time-Doppler chirp-varying (TDCV) filter is proposed for ground moving target indication and parameter estimation with the airborne SC-SAR system. The proposed method is easy to implement and mainly includes three steps. First, a traditional 2-D frequency range-Doppler algorithm is used to generate an original image. Second, the second-order range cell migration (RCM) phase term is partly compensated in the range frequency and azimuth time domain, and the rest of second-order RCM phase term is compensated in 2-D frequency domain. The whole processing, which is referred to as the TDCV approach, is employed to acquire a new TDCV image. Third, compared the original image with the new image, the clutter scatterers are nearly motionless while the moving targets are translated along the range direction due to their nonzero radial velocities. After the cancellation between two normalized images, the clutter background would be significantly suppressed since the two images generated by the same data. As a result, the moving targets can be indicated and the range difference of the moving target between two images can be exploited to estimate their radial velocities. The results obtained by applying the proposed method into a set of real SAR data are consistent with the analysis presented in this paper.