We consider short-pulse (SP) time-domain (TD) two-dimensional (2-D) scattering by moderately rough interfaces, which separate free space from a slightly lossy dielectric half-space, and are excited by one-dimensional (1-D) SP-TD aperture field distributions. This study extends to the SP-TD in our previous investigation of time-harmonic high frequency 2-D scattering of Gabor-based quasi-ray Gaussian beam fields excited by 1-D aperture field distributions in the presence of moderately rough dielectric interfaces (Galdi et al.). The proposed approach is based on the Kirchhoff physical optics (PO) approximation in conjunction with the Gabor-based quasi-ray narrow-waisted Gaussian pulsed-beam (PB) discretization (Galdi et al.), which is applied to the SP-induced equivalent magnetic surface currents on the interface that establish the TD reflected/transmitted fields. We show that, for well-collimated truncated SP incident fields, the PO-PB synthesis of the reflected/transmitted fields yields an approximate explicit physically appealing, numerically efficient asymptotic algorithm, with well-defined domains of validity based on the problem parameters. An extensive series of numerical experiments verifies the accuracy of our method by comparison with a rigorously-based numerical reference solution, and assesses its computational utility. The algorithm is intended for use as a rapid forward solver in SP-TD inverse scattering and imaging scenarios in the presence of moderately rough dielectric interfaces.