The fast modulation of lasers is a fundamental requirement for applications in optical communications, high-resolution spectroscopy and metrology. In the terahertz-frequency range, the quantum-cascade laser (QCL) is a high-power source with the potential for high-frequency modulation. However, conventional electronic modulation is limited fundamentally by parasitic device impedance, and so alternative physical processes must be exploited to modulate the QCL gain on ultrafast timescales. Here, we demonstrate an alternative mechanism to modulate the emission from a QCL device, whereby optically-generated acoustic phonon pulses are used to perturb the QCL bandstructure, enabling fast amplitude modulation that can be controlled using the QCL drive current or strain pulse amplitude, to a maximum modulation depth of 6% in our experiment. We show that this modulation can be explained using perturbation theory analysis. While the modulation rise-time was limited to ~800 ps by our measurement system, theoretical considerations suggest considerably faster modulation could be possible.