Abstract This paper proposes a fast algorithm for sparse spectral estimation to achieve super-resolution SAR (synthetic aperture radar) sparse imaging of complex targets. The fast algorithm used the Newton method and fast Fourier transform (FFT) based preconditioned conjugate gradient (PCG) method to estimate sparse spectral parameters efficiently. Finally, a sparse SAR imaging was obtained based on sparse spectral parameters. It can suppress imaging sidelobe clutter and speckle with a super-resolution. The experiments with simulated and real targets were used to demonstrate the computational efficiency and super-resolution sparse imaging capability of the proposed algorithm.