Context. Massive stars play crucial roles in galactic dynamics and chemical evolution. They are the most significant sources of ionizing UV radiation, their substantial mass-loss rates and explosions inject energy and enrich their surroundings, and their dynamical interactions eject stars and alter the evolution of stellar clusters. Consequently, the study of massive stars is essential for understanding various astrophysical phenomena, including galaxy chemical evolution, interstellar medium dynamics, gamma-ray bursts, and the reionization of the Universe. Key parameters influencing the evolution of massive stars include mass, mass-loss rate, chemical composition, and rotation. The orbits of spectroscopic binaries are particularly valuable because they provide constraints on stellar masses, and when combined with complementary data (e.g., photometry or interferometry), these masses can be fully determined. Aims. The OWN Survey was started two decades ago to study Galactic O- and WN- (hence the name) type southern spectroscopic binaries. In this paper we present the final results for single-lined (SB1) spectroscopic orbits. Methods. The OWN Survey carried out a long-term spectroscopic campaign to search for radial velocity variations indicative of orbital motion in a sample of southern Galactic O- and WN-type stars with high-resolution spectrographs in Argentina and Chile. The OWN spectra were later combined with high-resolution spectra from other sources and, in some cases, photometric time series to derive orbits and disentangled spectra, from which masses were constrained or determined and spectral classifications obtained. High-resolution optical spectra of 212 massive stars were obtained during the ∼20 years of the OWN project, and each target was observed at least three times. Results. Among the 212 stars, 144 exhibited radial-velocity variations greater than 15 km s−1. We present a complete and coherent compilation for the 23 systems with single-lined spectroscopic orbits identified in our sample. In Paper II we will perform a similar analysis for the systems with double-lined spectroscopic orbits.