The Astrophysics with Italian Replicating Technology Mirrors (ASTRI) Mini-Array is an international collaboration led by the Italian National Institute for Astrophysics (INAF) and devoted to imaging atmospheric Cherenkov light for very-high γ-ray astrophysics, detection of cosmic-rays, and stellar Hambury-Brown intensity interferometry. The project is deploying an array of nine dual-mirror aplanatic imaging atmospheric Cherenkov telescopes of 4-m class at the Teide Observatory on Tenerife in the Canary Islands. Based on SiPM sensors, the focal plane camera covers an unprecedented field of view of 10.5 deg in diameter. The array is most sensitive to γ-ray radiation above 1 up to 200 TeV, with an angular resolution of 3 arcmin, better than the current particle arrays, such as LHAASO and HAWC. We describe the overall software architecture of the ASTRI Mini-Array and the software engineering approach for its development. The software covers the entire life cycle of the Mini-Array, from scheduling to remote operations, data acquisition, and processing until data dissemination. The on-site control software allows remote array operations from different locations, including automated reactions to critical conditions. All data are collected every night, and the array trigger is managed post facto. The high-speed networking connection between the observatory site and the Data Center in Rome allows for ready data availability for stereoscopic event reconstruction, data processing, and almost real-time science products generation.