The generation of terahertz (THz) spin current is highly desired for enhancing the speed and efficiency of information manipulation by ultrafast spintronics. We develop an analytical model to investigate the coherent and pure spin-current pulses generated by spin pumping in a canted insulating antiferromagnet (AFM) system with Dzyaloshinskii-Moriya interaction (DMI). As expected, the spin pumping signal is significant at resonance. However, we also predict efficient THz spin-current transient with a magnitude comparable with that of its resonant counterparts in the much higher off-resonance THz frequency regime. Its ultrafast temporal change compensates for the suppression of the dynamic magnetization amplitudes at off-resonance. In addition, a stronger DMI field enhances the amplitude of the spin-current transient. Our results represent an efficient approach to generate coherent THz spin-current pulses based on canted AFM heterostructures without requiring a static magnetic field.