We present a straightforward and computationally cheap method to obtain the phonon-assisted photocurrent in large-scale devices from first-principles transport calculations. The photocurrent is calculated using the nonequilibrium Green's function with light-matter interaction from the first-order Born approximation, while electron-phonon coupling (EPC) is included through special thermal displacements. We apply the method to a silicon solar-cell device and demonstrate the impact of including EPC in order to properly describe the current due to the indirect band-to-band transitions. The first-principles results are successfully compared to experimental measurements of the temperature and light-intensity dependence of the open-circuit voltage of a silicon photovoltaic module. Our calculations illustrate the pivotal role played by EPC in photocurrent modeling to avoid underestimation of the open-circuit voltage, shortcircuit current, and maximum power. This work represents a recipe for computational characterization of future photovoltaic devices including the combined effects of light-matter interaction, phonon-assisted tunneling, and the device potential at finite bias from the level of first-principles simulations.