Recently, accurate parameter estimation of the damped complex exponential plays an increasingly important role in the field of precise measurement. However, the estimation variance of interpolation-based algorithms for the parameter estimation cannot be asymptotic to the Crámer-Rao lower bound (CRLB). This paper originally proposes a generalized, fast, and the accurate two-iteration estimator (TIE) based on the discrete Fourier transform (DFT). It can be operated by an arbitrary window (symmetric or asymmetric window). Theoretical estimation variances of the frequency and the damping factor for arbitrary windows are derived, respectively. Furthermore, extensive computer simulations are performed to compare the performance of the TIE with other state-of-the-art algorithms in the literature. The results support the theoretical findings and verify that high-accuracy parameter estimation can be ensured by the proposed algorithm. More importantly, the estimation variances returned by the TIE with the rectangle window exactly track the CRLB for a damped single tone.