The Great Nebula in Carina (NGC 3372) is the best target to study in detail the process of violent massive star formation and the resulting feedback effects of cloud dispersal and triggered star formation. While the population of massive stars is rather well studied, the associated low-mass stellar population was largely unknown up to now. The near-infrared study in this paper builds on the results of the Chandra Carina Complex Project, that detected 14368 X-ray sources in the 1.4deg^2^ survey region, an automatic source classification study that classified 10714 of these X-ray sources as very likely young stars in Carina, and an analysis of the clustering properties of the X-ray-selected Carina members. In order to determine physical properties of the X-ray-selected stars, most of which were previously unstudied, we used HAWK-I at the ESO Very Large Telescope to conduct a very deep near-IR survey with subarcsecond angular resolution, covering an area of about 1280arcmin^2^. The HAWK-I images reveal more than 600000 individual infrared sources, whereby objects as faint as J~23, H~22, and Ks~21 are detected at signal-to-noise ratio (S/N)>=3. While less than half of the Chandra X-ray sources have counterparts in the Two Micron All Sky Survey catalog, the ~5mag deeper HAWK-I data reveal infrared counterparts to 6636 (=88.8%) of the 7472 Chandra X-ray sources in the HAWK-I field. We analyze near-infrared color-color and color-magnitude diagrams to derive information about the extinctions, infrared excesses (as tracers for circumstellar disks), ages, and masses of the X-ray-selected objects.