Knowledge about the large-scale distribution of galaxies is far from complete in the Zone of Avoidance, which is mostly due to high interstellar extinction and to source confusion at lower Galactic latitudes. Past near-infrared (NIR) surveys, such as the Two Micron All Sky Survey (2MASS), have shown the power of probing large-scale structure at these latitudes. Our aim is to map the galaxy distribution across the Southern Galactic plane using the VISTA Variables in the Via Lactea Survey (VVV), which reach 2 to 4 magnitudes deeper than 2MASS. We used SExtractor + PSFEx to identify extended objects and to measure their sizes, the light concentration index, magnitudes, and colours. Morphological and colour constraints and visual inspection were used to confirm galaxies. We present the resulting VVV NIR Galaxy Catalogue of 5563 visually confirmed galaxies, of which only 45 were previously known. This is the largest catalogue of galaxies towards the Galactic plane, with 99% of these galaxies being new discoveries. We found that the galaxy density distribution closely resembled the distribution of low interstellar extinction of the existing NIR maps. We also present a description of the 185 2MASS extended sources observed in the region, of which 16% of these objects had no previous description, which we have now classified. We conclude that interstellar extinction and stellar density are the main limitations for the detection of background galaxies in the Zone of Avoidance. The VVV NIR Galaxy Catalogue is a new data set providing information for extragalactic studies in the Galactic plane.